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7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 118
Discrete Mathematics Structures
6 1391
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 218
983094983089983091983097983089
Predicate Logic - everybody loves somebody
Proposition YES or NO3 + 2 = 5
X + 2 = 5
X + 2 = 5 for any choice of X in 1 2 3X + 2 = 5 for some X in 1 2 3
YES
NO
YES
YES
- 983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 318
983094983089983091983097983089
Predicate Logic - everybody loves somebody
Alicia eats pizza at least once a weekGarrett eats pizza at least once a weekAllison eats pizza at least once a week
Gregg eats pizza at least once a weekRyan eats pizza at least once a weekMeera eats pizza at least once a week
Ariel eats pizza at least once a week
hellip - 983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 418
983094983089983091983097983089
Predicates
Alicia eats pizza at least once a week
DefineEP(x) = ldquox eats pizza at least once a weekrdquoUniverse of Discourse - x is a student in DM
A predicate or propositional function is a functionthat takes some variable(s) as arguments andreturns True or False
Note that EP(x) is not a proposition EP( Ariel ) is
hellip
- 983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 518
983094983089983091983097983089
Predicates
Suppose Q(xy) = ldquox gt y rdquo
Proposition YES or NO
Q(xy) Q( 34 ) Q(x9 )
NO
YES
NO
Predicate YES or NO
Q(xy)
Q( 34 )
Q(x9 )
YES
NO
YES
- 983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 618
983094983089983091983097983089
Predicates - the universal quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx B(x) = ldquox is carrying a backpackrdquo x is set of DM students
The universal quantifier of P(x) is the proposition
ldquoP(x) is true for all x in the universe of discourserdquo
We write it forallx P(x) and say ldquofor all x P(x)rdquo
forallx P(x) is TRUE if P(x) is true for every single xforallx P(x) is FALSE if there is an x for which P(x) is false
forallx B(x)
- 983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 218
983094983089983091983097983089
Predicate Logic - everybody loves somebody
Proposition YES or NO3 + 2 = 5
X + 2 = 5
X + 2 = 5 for any choice of X in 1 2 3X + 2 = 5 for some X in 1 2 3
YES
NO
YES
YES
- 983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 318
983094983089983091983097983089
Predicate Logic - everybody loves somebody
Alicia eats pizza at least once a weekGarrett eats pizza at least once a weekAllison eats pizza at least once a week
Gregg eats pizza at least once a weekRyan eats pizza at least once a weekMeera eats pizza at least once a week
Ariel eats pizza at least once a week
hellip - 983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 418
983094983089983091983097983089
Predicates
Alicia eats pizza at least once a week
DefineEP(x) = ldquox eats pizza at least once a weekrdquoUniverse of Discourse - x is a student in DM
A predicate or propositional function is a functionthat takes some variable(s) as arguments andreturns True or False
Note that EP(x) is not a proposition EP( Ariel ) is
hellip
- 983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 518
983094983089983091983097983089
Predicates
Suppose Q(xy) = ldquox gt y rdquo
Proposition YES or NO
Q(xy) Q( 34 ) Q(x9 )
NO
YES
NO
Predicate YES or NO
Q(xy)
Q( 34 )
Q(x9 )
YES
NO
YES
- 983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 618
983094983089983091983097983089
Predicates - the universal quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx B(x) = ldquox is carrying a backpackrdquo x is set of DM students
The universal quantifier of P(x) is the proposition
ldquoP(x) is true for all x in the universe of discourserdquo
We write it forallx P(x) and say ldquofor all x P(x)rdquo
forallx P(x) is TRUE if P(x) is true for every single xforallx P(x) is FALSE if there is an x for which P(x) is false
forallx B(x)
- 983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 318
983094983089983091983097983089
Predicate Logic - everybody loves somebody
Alicia eats pizza at least once a weekGarrett eats pizza at least once a weekAllison eats pizza at least once a week
Gregg eats pizza at least once a weekRyan eats pizza at least once a weekMeera eats pizza at least once a week
Ariel eats pizza at least once a week
hellip - 983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 418
983094983089983091983097983089
Predicates
Alicia eats pizza at least once a week
DefineEP(x) = ldquox eats pizza at least once a weekrdquoUniverse of Discourse - x is a student in DM
A predicate or propositional function is a functionthat takes some variable(s) as arguments andreturns True or False
Note that EP(x) is not a proposition EP( Ariel ) is
hellip
- 983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 518
983094983089983091983097983089
Predicates
Suppose Q(xy) = ldquox gt y rdquo
Proposition YES or NO
Q(xy) Q( 34 ) Q(x9 )
NO
YES
NO
Predicate YES or NO
Q(xy)
Q( 34 )
Q(x9 )
YES
NO
YES
- 983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 618
983094983089983091983097983089
Predicates - the universal quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx B(x) = ldquox is carrying a backpackrdquo x is set of DM students
The universal quantifier of P(x) is the proposition
ldquoP(x) is true for all x in the universe of discourserdquo
We write it forallx P(x) and say ldquofor all x P(x)rdquo
forallx P(x) is TRUE if P(x) is true for every single xforallx P(x) is FALSE if there is an x for which P(x) is false
forallx B(x)
- 983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 418
983094983089983091983097983089
Predicates
Alicia eats pizza at least once a week
DefineEP(x) = ldquox eats pizza at least once a weekrdquoUniverse of Discourse - x is a student in DM
A predicate or propositional function is a functionthat takes some variable(s) as arguments andreturns True or False
Note that EP(x) is not a proposition EP( Ariel ) is
hellip
- 983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 518
983094983089983091983097983089
Predicates
Suppose Q(xy) = ldquox gt y rdquo
Proposition YES or NO
Q(xy) Q( 34 ) Q(x9 )
NO
YES
NO
Predicate YES or NO
Q(xy)
Q( 34 )
Q(x9 )
YES
NO
YES
- 983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 618
983094983089983091983097983089
Predicates - the universal quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx B(x) = ldquox is carrying a backpackrdquo x is set of DM students
The universal quantifier of P(x) is the proposition
ldquoP(x) is true for all x in the universe of discourserdquo
We write it forallx P(x) and say ldquofor all x P(x)rdquo
forallx P(x) is TRUE if P(x) is true for every single xforallx P(x) is FALSE if there is an x for which P(x) is false
forallx B(x)
- 983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 518
983094983089983091983097983089
Predicates
Suppose Q(xy) = ldquox gt y rdquo
Proposition YES or NO
Q(xy) Q( 34 ) Q(x9 )
NO
YES
NO
Predicate YES or NO
Q(xy)
Q( 34 )
Q(x9 )
YES
NO
YES
- 983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 618
983094983089983091983097983089
Predicates - the universal quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx B(x) = ldquox is carrying a backpackrdquo x is set of DM students
The universal quantifier of P(x) is the proposition
ldquoP(x) is true for all x in the universe of discourserdquo
We write it forallx P(x) and say ldquofor all x P(x)rdquo
forallx P(x) is TRUE if P(x) is true for every single xforallx P(x) is FALSE if there is an x for which P(x) is false
forallx B(x)
- 983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 618
983094983089983091983097983089
Predicates - the universal quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx B(x) = ldquox is carrying a backpackrdquo x is set of DM students
The universal quantifier of P(x) is the proposition
ldquoP(x) is true for all x in the universe of discourserdquo
We write it forallx P(x) and say ldquofor all x P(x)rdquo
forallx P(x) is TRUE if P(x) is true for every single xforallx P(x) is FALSE if there is an x for which P(x) is false
forallx B(x)
- 983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 718
983094983089983091983097983089
Predicates - the universal quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) forallx (Y(x) rarr B(x))b) forallx (Y(x) or L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of discourse
is people in this room
- 983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 818
983094983089983091983097983089
Predicates - the existential quantifier
Another way of changing a predicate into a proposition
Suppose P(x) is a predicate on some universe of discourseEx C(x) = ldquox has a carrdquo x is set of DM students
The existential quantifier of P(x) is the proposition
ldquoP(x) is true for some x in the universe of discourserdquo
We write it existx P(x) and say ldquofor some x P(x)rdquo
existx P(x) is TRUE if there is an x for which P(x) is trueexistx P(x) is FALSE if P(x) is false for every single x
existx C(x)
- 983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 918
983094983089983091983097983089
Predicates - the existential quantifier
B(x) = ldquox is wearing sneakersrdquo
L(x) = ldquox is at least 21 years oldrdquoY(x)= ldquox is less than 24 years oldrdquo
Are either of these propositions true
a) existx B(x)b) existx (Y(x) and L(x))
A only a is true
B only b is true
C both are true
D neither is true
Universe of
discourse is peoplein this room
- 983097
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1018
983094983089983091983097983089
Predicates - more examples
Universe of discourse
is all creatures
L(x) = ldquox is a lionrdquo
F(x) = ldquox is fiercerdquoC(x) = ldquox drinks coffeerdquo
All lions are fierce
Some lions donrsquot drink coffee
Some fierce creatures donrsquot drink coffee
forallx (L(x) rarr F(x))
existx (L(x) and notC(x))
existx (F(x) and notC(x))
- 983089983088
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1118
983094983089983091983097983089
Predicates - more examples
Universe of discourseis all creatures
B(x) = ldquox is a hummingbirdrdquo
L(x) = ldquox is a large birdrdquoM(x) = ldquox lives on mountainrdquoR(x) = ldquox is richly coloredrdquo
All hummingbirds are richly colored
No large birds live on mountain
Birds that do not live on mountain are dully colored
forallx (B(x) rarr R(x))
notexistx (L(x) and M(x))
forallx (notM(x) rarr notR(x))
- 983089983089
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1218
983094983089983091983097983089
Predicates - quantifier negation
Not all large birds live on mountain
forallx P(x) means ldquoP(x) is true for every xrdquoWhat about notforallx P(x)
Not [ldquoP(x) is true for every xrdquo]
ldquoThere is an x for which P(x) is not truerdquoexistx notP(x)
Sonotforallnotforallnotforallnotforall
x P(x) is the same asexistexistexistexist
xnotnotnotnot
P(x)
notforallx (L(x) rarr M(x))
existx not(L(x) rarr M(x))
- 983089983090
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1318
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
existx P(x) means ldquoP(x) is true for some xrdquoWhat about notexistx P(x)
Not [ldquoP(x) is true for some xrdquo]
ldquoP(x) is not true for all xrdquoforallx notP(x)
Sonotexistnotexistnotexistnotexist
x P(x) is the same asforallforallforallforall
xnotnotnotnot
P(x)
notexistx (L(x) and M(x))
forallx not(L(x) and M(x))
- 983089983091
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1418
983094983089983091983097983089
Predicates - quantifier negation
So notforallx P(x) is the same as existx notP(x)So notexistx P(x) is the same as forallx notP(x)
General rule to negate a quantifier movenegation to the right changing quantifiers as
you go
- 983089983092
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1518
983094983089983091983097983089
Predicates - quantifier negation
No large birds live on Mountain
notexistx (L(x) and M(x)) equiv forallx not(L(x) and M(x)) Negationrule
equiv forall
x (not
L(x)or not
M(x)) DeMorganrsquosequiv forallx (L(x) rarr notM(x)) Subst for rarr
Whatrsquos wrong with thisproof
- 983089983093
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1618
983094983089983091983097983089
Predicates - free and bound variables
A variable is bound if it is known or quantified
Otherwise it is free
ExamplesP(x) x is freeP(5) x is bound to 5forallx P(x) x is bound by quantifier
Reminder in a
proposition allvariables must be
bound
- 983089983094
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1718
983094983089983091983097983089
Predicates - multiple quantifiers
To bind many variables use many quantifiers
Example P(xy) = ldquox gt yrdquoforallx P(xy)forallxforall y P(xy)forallxexist y P(xy)forallx P(x3)
a) True proposition
b) False proposition
c) Not a proposition
d) No clue
c)
b)
a)
b)
- 983089983095
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096
7252019 Discrete Mathematics Structures Slide 2
httpslidepdfcomreaderfulldiscrete-mathematics-structures-slide-2 1818
Predicates - the meaning of multiple
quantifiers
forallxforall y P(xy)
existxexist y P(xy)
forallxexist y P(xy)
existxforall y P(xy)
983094983089983091983097983089
P(xy) true for all x y pairs
For every value of x we can find a (possibly different)
y so that P(xy) is true
P(xy) true for at least one x y pair
There is at least one x for which P(xy)
is always true
quantification order is notcommutative
Suppose P(xy) = ldquoxrsquos favorite class is yrdquo
- 983089983096