Chapter 2: The Logic of QuantifiedStatements

January 14, 2008

Outline

1 2.1 Introduction to Predicates and Quantified Statements I

2 2.2 Introduction to Predicates and Quantified Statements II

3 2.3 Statements Containing Multiple Quantifiers

Consider the following argument

All students in Electrical Engineering have to take MTH314.Tarik is a student in Electrical Engineering.

∴ Tarik has to take MTH314.

This seems to be a valid argument from the point of view oflogic. However, we cannot show that using the methods ofChapter 1, since the first statement cannot be translated usingonly the symbols ∼, ∧, ∨, →, or ↔.

Basic Facts and Notation For Sets

• A set is some collection of objects, which we call elementsof the set. For example,

A = { blue , red , yellow }

andB = {0, 2, 5, 13,−4}

are examples of sets.• To write the fact that an object x is an element of a set A,

we writex ∈ A

whilex 6∈ A

will denote the fact that x is not an element of A.

• We will use the following notation for familiar sets ofnumbers:

1 R for the set of all real numbers2 Z for the set of all integers3 Q for the set of all rational numbers (quotients of integers)

• By putting + or - in the superscript, we indicate the set ofpositive or negative numbers; e.g

R+

will stand for all positive real numbers, etc.• Instead of writing Z+, the set of all positive integers is often

called the set of natural numbers and the notation for it isN.

Predicates

• We have seen in Chapter 1, that a sentence

x2 + x > 2

is not a statement since its truth (or falsity) depends onwhat value we assign to the variable x .

• Similarly, the sentence

She is a student at Ryerson.

is not a statement either, since we have to know who “she”refers to in order to determine the truth value.

• Such sentences are called predicates. They are notstatements unless we interpret the variables in them asparticular elements of some prescribed set. After that, theybecome true or false statements.

DefinitionA predicate is a sentence that contains a finite number ofvariables and becomes a statement once these variables areassigned some specific values. The domain of a predicatevariable is the set of all values that can be substituted for thevariable.

• For example, for the predicate P(x)

x2 + x > 2

the domain can be any set of numbers (R, Z, . . . ) in whichits operations (+,·,. . . ) make sense.

• A predicate could involve any number of variables; e.g.Q(x , y) is the sentence

x is divisible by y3

ExampleConsider the predicate P(x)

x2 + x > 2

with the domain R. For what x ∈ R is this predicate true?

• For x = 2,P(2) : 22 + 2 > 2, true

• For x = −13 ,

P(−13) : (−1

3)2 + (−1

3) > 1, false

• It can be shown, using some basic facts about quadraticequations that P(x) is true provided

x < −2 or x > 1

DefinitionIf P(x) is a predicate whose truth domain is D, the truth set ofP(x) is the set of all elements of D which make P(x) true whenthey are substituted for x . We write the truth set for P(x) as

{x ∈ D|P(x)}

which we read as “the set of all x in D such that P(x) is true”.

ExampleConsider the predicate Q(n) stating

n divides 5

(a) If the truth domain is Z, the truth set of this predicate is

{−5,−1, 1, 5}

(b) If the truth domain is the set of all positive integers Z+,then the truth set will be

{1, 5}

• We see, based on this example, that for the samepredicate, the truth set may change when we change thetruth domain (the set of possible values of the variable)

Universal Quantifier

• One of the ways in which we can turn a predicate P(x) intoa statement is, for example, to state the following

For all x , P(x).

• We write that as∀x , P(x)

• The symbol ∀ means “for all” (“for any”, “for each”) and iscalled the universal quantifier.

• For example, the sentenceAll students in Electrical Engineering must take MTH314.

can be written as∀x ∈ E , P(x)

where E is the set of all students in Electrical Engineering,and P(x) is the predicate “x must take MTH314”.

DefinitionSuppose Q(x) is a predicate and D is the truth domain for x . Auniversal statement is a statement of the form

∀x ∈ D, Q(x).

It is true if, and only if, Q(x) is true for every x in D. If there isan x in D for which Q(x) is false, this universal statement willbe false.The value of x for which Q(x) is false is called acounterexample for the universal statement.

ExampleLet D = {−5, 8, 9, 11} and consider the statement

∀x ∈ D, x2 + x > 2

Show that this statement is true.Solution: We check that

x2 + x > 2

is true for all x ∈ D. (This is called the method of exhaustion)

(−5)2 + (−5) > 2

82 + 8 > 2

92 + 9 > 2

112 + 11 > 2

∀x ∈ D, x2 + x > 2

is true

ExampleNow, show that

∀x ∈ R, x2 + x > 2

is false.Solution: It suffices to find one value of x ∈ R for which it is notthe case that

x2 + x > 2

In other words, we are looking for a counterexample in R.One such counterexample is, e.g. x = −1 since

(−1)2 + (−1) 6> 2

Therefore,∀x ∈ R, x2 + x > 2

is false.

Existential Quantifier

• Another way to turn a predicate Q(x) into a sentence is toassert that

“There exists x such that Q(x).”• The notation for this is

∃x , Q(x)

• ∃ is the existential quantifier and its meaning is “thereexists”, “there is a”, “for some”, “there is at least one”, etc.

DefinitionLet Q(x) be a predicate and D the truth domain of x . Anexistential statement is a statement of the form

∃x ∈ D such that Q(x)

It is true if, and only if, Q(x) is true for at least one element x ofD. It will be false if, Q(x) is false for all x ∈ D.

ExampleShow that the statement

∃n ∈ Z such that n2 = n

is true.Solution: It suffices to find an integer n such that

n2 = n

Obviously, one such integer is

n = 1.

ExampleGiven the truth domain

A = {2,−7, 3, 5}

show that the existential statement

∃n ∈ A such that n2 = n

is false.Solution: Clearly,

22 6= 2

(−7)2 6= −7

32 6= 3

52 6= 5

Since there is no element n ∈ A for which n2 = n, thisexistential statement is false.

• Given a statement involving quantifiers, it is generallypossible to translate it into English using in a variety ofways:

Example

∀x ∈ R, x2 ≥ 0.

This can be translated as:• Every real number has a non-negative square.• x has a non-negative square, for any real number x .• No real number can have a non-negative square, etc.

ExamplesTranslate the following statements into formal logic statements,using quantifiers and variables:(a) The sum of angles of every rectangle is 360◦.(b) Some integers are divisible by 6.(c) No real number has -4 as its square.

Solution:(a) ∀x ∈ R, sum of angles of x is 360◦.

(R-set of all rectangles)(b) ∃n ∈ Z such that n is divisible by 6.(c) ∀x ∈ R, x2 6= −4.

Universal Conditional Statements• One of the most common types of universal statements

encountered in mathematics is the universal conditionalstatement

∀x , if P(x) then Q(x).

ExampleConsider the statement

∀x ∈ R, if x ≥ 3 then x2 ≥ 9.

which can be translated as

If a real number is at least 3, then its square is at least 9.

or

The square of any real number, which is at least 3, is at least 9.

ExamplesRewrite each of the following statements as a universalconditional statement:(a) If an integer is divisible by 4, then it is divisible by 2.(b) No bird is a mammal.

Solution:(a) ∀n ∈ Z, if n is divisible by 4, then n is divisible by 2.

(b) ∀x , if x is a bird, then x is not a mammal.

Equivalent Forms of QuantifiedStatements

• Consider the statement∀x , if x ∈ Z, then x ∈ Q.

(Every integer is a rational number)• This statement can be rewritten as

∀x ∈ Z, x ∈ Q• What we did here was to restrict the truth domain to

integers.• Conversely, if D is the truth domain for the predicate P(x),

the statement∀x ∈ D, P(x)

can be rewritten as∀x , if x ∈ D, then P(x)

ExampleRewrite the statement

The square of any even integer is even.

as a universal logical statement in two different ways.Solution:• ∀n, if n is even, then n2 is even.• ∀ even n, n2 is even.

ExampleA prime number is an integer greater than 1 whose onlypositive factors are 1 and the integer itself.Let Prime(n) stand for the predicate “n is prime”, and letEven(n) denote the predicate “n is even”.Consider the statement

There is an integer that is prime and even.

Rewrite this statement as an existential statement using thepredicates Prime(n) and Even(n).Solution:• ∃n such that Prime(n) ∧ Even(n).• ∃ an even integer n such that Prime(n).• ∃ a prime integer n such that Even(n).

Implicit Quantification

• It is quite common in mathematics that statements whichseemingly are not quantified contain quantifiers implicitly

Example

1 If x ≥ 3 then x2 ≥ 9.

∀x ∈ R, x ≥ 3 → x2 ≥ 9

2 12 is a product of an even and an odd integer.

∃ odd m and even n, such that 12 = m · n

• The statement

∀x ∈ R, x ≥ 3 → x2 ≥ 9

can also be written in the following way (which is quitecommon in mathematics):

x ≥ 3 =⇒ x2 ≥ 9

DefinitionIf P(x) and Q(x) are predicates which have the same domainD, the notation

P(x) =⇒ Q(x)

means that every element x ∈ D which is in the truth set forP(x) is also in the truth set of Q(x). Equivalently,

∀x , P(x) → Q(x)

The notationP(x) ⇐⇒ Q(x)

means that the universal statement

∀x , P(x) ↔ Q(x)

is true; i.e. that P(x) and Q(x) have the same truth sets.

ExampleSuppose Q(n), R(n), and S(n) are the following predicates

Q(n): “n is a factor of 8.”R(n): “n is a factor of 4.”S(n): “n < 5 and n 6= 3.”

whose domain is Z+. use =⇒ and ⇐⇒ to find the relationshipbetween predicates Q(n), R(n) and S(n).

Solution: The truth sets in Z+ for these three statements are

Q(n) :{1, 2, 4, 8}R(n) :{1, 2, 4}S(n) :{1, 2, 4}

We see that:

R(n) ⇐⇒ S(n)

R(n) =⇒ Q(n)

S(n) =⇒ Q(n)

Negations of Quantified Statements

• Consider the following statement:“All engineering students have to take MTH 314.”

• If we want to negate this statement, intuitively, we wouldphrase it in the following way:“There are some engineering students which do not have

to take MTH 314.”• Note that we cannot negate the original statement by

saying:“No engineering students have to take MTH 314.”

Theorem(Negation of a Universal Statement) The negation of astatement

∀x in D, P(x)

is logically equivalent to a statement of the form

∃x in D such that ∼ P(x)

We can write that as

∼ (∀x ∈ D, P(x)) ≡ ∃x ∈ D such that ∼ P(x)

• If we were to negate the existential statement

“Some birds are mammals.”

the correct logical reasoning would indicate that it shouldsay:

“No birds are mammals.”

instead of

“Some birds are not mammals.”

Theorem(Negation of an Existential Statement) The negation of astatement

∃x in D such that P(x)

is logically equivalent to a statement of the form

∀x in D, ∼ P(x)

We can write that as

∼ (∃x ∈ D such that P(x)) ≡ ∀x ∈ D, ∼ P(x)

ExamplesWrite the formal negations of the following statements:(a) ∀ prime n, n is odd.

(b) ∃ a continuous function f on [a, b] such that∫ b

a f (x)dx doesnot exist.

(c) No integers are divisible by 6.

Solution:(a) ∃ a prime n such that n is not odd.

(b) ∀ continuous function f on [a, b],∫ b

a f (x)dx exists.(c) This statement is equivalent to: “∀ integer n, n is not

divisible by 6.”So, its negation is equivalent to: “∃ an integer n such that nis divisible by 6.”

Negations of Universal ConditionalStatements

• Suppose we want to negate a universal conditionalstatement

∀x , P(x) → Q(x)

• Then, based on what we have just seen about negations ofuniversal statements, we see that

∼ (∀x , P(x) → Q(x)) ≡ ∃x such that ∼ (P(x) → Q(x))

• However,

∼ (P(x) → Q(x)) ≡ P(x)∧ ∼ Q(x)

• So,

∼ (∀x , P(x) → Q(x)) ≡ ∃x such that (P(x)∧ ∼ Q(x)).

• We have just proved the equivalence

∼ (∀x , P(x) → Q(x)) ≡ ∃x such that (P(x)∧ ∼ Q(x))

ExamplesWrite formal negations for the following statements:(a) ∀ integer n, if n is even, then n is divisible by 4.(b) If a computer program P compiles without any error

messages, then P is correct.

Solution:(a) ∃ an integer n such that n is even and n is not divisible by 4.(b) ∃ a computer program P such that P compiles without any

error messages and P is not correct.

Relation Among ∀, ∃, ∧, and ∨

• Suppose we are looking at quantified statements whichrefer to a finite domain

D = {x1, x2, . . . , xn}

• Saying that the statement

∀x ∈ D, P(x)

is true, is equivalent to saying that P(x) is true for everyxi ∈ D:

P(x1) ∧ P(x2) ∧ . . . ∧ P(xn)

ExampleSuppose

D = {1,−1}

and P(x) is the predicate x2 = 1. Then

∀x ∈ D, P(x)

is true sinceP(1) ∧ P(−1)

is true.

• On the other hand,

∃x ∈ D such that P(x)

is equivalent to the statement

P(x1) ∨ P(x2) ∨ . . . ∨ P(xn)

ExampleIn the same domain D = {1,−1} consider the predicate Q(x)stating that x2 = x . In that case,

∃x ∈ D such that Q(x)

is true sinceQ(1) ∨Q(−1)

• Remark: If we have a universal statement

∀x ∈ D, if P(x) then Q(x)

then this statement is vacuously true if, and only if, P(x) isfalse for every x ∈ D.

ExampleFor instance, the statement

∀x ∈ R, if x 6= x then x > 0

is vacuously true, because the hypothesis of the predicate

x 6= x

is false for every x ∈ R.

Variants of Universal ConditionalStatements

DefinitionConsider a universal conditional statement

∀x ∈ D, if P(x) then Q(x)

1 Its contrapositive is the statement

∀x ∈ D, if ∼ Q(x) then ∼ P(x)

2 Its converse is

∀x ∈ D, if Q(x) then P(x)

3 Its inverse is

∀x ∈ D, if ∼ P(x) then ∼ Q(x)

ExampleGiven the statement

∀x ∈ R, if x > 2, then 3x + 2 > 8

write its contrapositive, converse and inverse statements.Solution:• Contrapositive:

∀x ∈ R, if 3x + 2 6> 8, then x 6> 2

• Converse:

∀x ∈ R, if 3x + 2 > 8, then x > 2

• Inverse:

∀x ∈ R, if x 6> 2, then 3x + 2 6> 8.

• The usual equivalences (and non-equivalences) are alsotrue for universal conditional statements; namely:

conditional statement ≡ its contrapositiveconditional statement 6≡ its converseconditional statement 6≡ its inverse

converse ≡ inverse

Necessary and Sufficient Conditions;Only If

Definition

1. “∀x , P(x) is a sufficient condition for Q(x)” means

∀x , if P(x) then Q(x)

2. “∀x , P(x) is a necessary condition for Q(x)” means

∀x , if ∼ P(x) then ∼ Q(x)

or, equivalently,

∀x , if Q(x) then P(x)

Definition(Only If) “∀x , P(x) only if Q(x)” means

∀x , if ∼ Q(x) then ∼ P(x)

or, equivalently,∀x , if P(x) then Q(x)

ExamplesRewrite each of the following statements as a universalconditional statement:(a) Squareness is a sufficient condition for rectangularity.(b) Being at least 18 years old is a necessary condition for

eligibility to vote in national elections.(c) A product of two integers is odd only if both of them are

odd.

Solution:(a) ∀x , if x is a square, then x is a rectangle.(b) ∀ people x , if x is eligible to vote in national elections, then

x is at least 18 years old.(c)

∀m, n ∈ Z, if mn is odd , then m, n are both odd

Statements Containing MultipleQuantifiers

Consider the following two statements about real numbers:

(a)∀x ∈ R,∃y ∈ R, such that x < y

(b)∃y ∈ R, such that ∀x ∈ R, x < y

These two statements say the following:• For every real number x , there is a larger real number y

(True.)• There exists a real number y such that it is larger than any

real number x . (False.)

Conclusion: We see that by reversing the order of quantifiers ina statement which involves two or more quantifiers, thestatement may change considerably.

Question:Given a statement of the form

∀x ∈ D,∃y ∈ E such that P(x , y)

or∃x ∈ D such that ∀y ∈ E , P(x , y)

how do we verify whether such a statement is true or false?

Suppose we want to check the truth value of the statement

∀x ∈ D,∃y ∈ E such that P(x , y)

We can think of it as a game we play with another player (TheSpoiler) which works in the following way:

1 The Spoiler picks an arbitrary element x in D.2 You respond by trying to find an element y in E for which

P(x , y) is true.3 If you have no answer for The Spoiler’s move, the

statement is false; if you can match any choice of x in TheSpoiler’s move with a good choice of y , you win and thestatement is true.

ExampleIn the All Here Tilomino world, we want to check the truth valueof the statement:

For every circle x , there is a triangle y which is east of x .

Solution:• If The Spoiler picks the unnamed circle, we pick either the

unnamed triangle in the same row or b.• If The Spoiler picks g, we pick the triangle b.• For The Spoiler’s choice of d , we can respond by choosing

b.Since we have a winning strategy for the game, the statementis true.

Suppose we want to check the truth value of the statement

∃x ∈ D, such that ∀y ∈ E , P(x , y)

We can think of it as a game we play with another player (TheSpoiler) which works in the following way:

1 We pick one fixed element x of D, which we are notallowed to change until the end of the game.

2 The Spoiler tries to challenge our choice of x by listing allpossible choices of y from E and hoping that, for at leastone of them, he/she will show that P(x , y) fails. (In themeantime, we twiddle our thumbs and hope for the best...)

3 If the spoiler doesn’t succeed in challenging our choice ofx , we have won the game and the statement is true; if theymanage to find a y in E which will disprove P(x , y), welose the game and the statement is false.

ExampleAgain, in the All here Tilomino world, we want to check the truthvalue of the statement

There is a circle x such that, for all squares y , x is north of y .

Solution:• Suppose we choose the circle in row 2 of the grid.• The Spoiler lists e; it does not disprove the statement.• The Spoiler then tries f ; still, our choice of the circle works.• Finally, The Spoiler is left with the last square, a; our

choice is still a good one.The Spoiler’s attempts to challenge our choice of x failed. So,we have won the game and the statement is true in thisTilomino world.

ExampleThe reciprocal of a real number a is a real number b such thatab = 1.The following two statements are true. Rewrite them usingquantifiers and variables.(a) Every nonzero real number has a reciprocal.(b) There is a real number with no reciprocal.

Solution:(a) ∀ nonzero real number x , ∃ a real number y such that

xy = 1(b) ∃ a real number z such that ∀ real numbers u, zu 6= 1.

Examples

(a) The statement

∃x ∈ Z+ such that ∀y ∈ Z+, x ≤ y

is true since it expresses the fact that there is the smallestpositive integer (which is 1).

(b) Now, if we change the domains of x and y to R+, positivereal numbers, we have

∃x ∈ R+ such that ∀y ∈ R+, x ≤ y

which is false, since, no matter how small x is, there is asmaller positive real number (e.g. x/2)

Negations of Statements with MultipleQuantifiers

• Suppose we want to negate the statement

∀x ∈ D,∃y ∈ E such that P(x , y)

• Then, using what we have already learned about negatingquantified statements, we have

∼ (∀x ∈ D,∃y ∈ E such that P(x , y))

≡ ∃x ∈ D such that ∼ (∃y ∈ E such that P(x , y))

≡ ∃x ∈ D such that ∀y ∈ E , ∼ P(x , y)

• Similarly, we can show that the statement

∼ (∃x ∈ D such that ∀y ∈ E , P(x , y))

is equivalent to the statement

∀x ∈ D,∃y ∈ E such that ∼ P(x , y)

ExampleIn the All Here Tilomino world, write a negation for each of thefollowing statements and determine their truth values of thesenegations:(a) For all circles x , there is a square y such that x and y have

the same size.(b) There is a square x such that, for every triangle y , x is

west of y .

Solution:(a) There is a circle x such that, for every square y , x and y

are not of the same size. (This statement is false)(b) For every square x , there exists a triangle y , so that x is

not west of y . (This statement is true)

Order of Quantifiers• As we have seen earlier, the order of quantifiers matters.

E.g., the following two statements are not equivalent:

∀x ∈ R∃y ∈ R such that x ≤ y

and∃y ∈ R such that ∀x ∈ R, x ≤ y

• However, if the quantifiers are identical, then the order isirrelevant. For example,

∀x ∈ R and ∀y ∈ R, x · y = y · x

is logically equivalent to

∀y ∈ R and ∀x ∈ R, x · y = y · x .

• This equivalence is also true in the case when there aretwo existential quantifiers.

Remarks: The following is also true1 “∀x ∈ D, P(x)” can be written as

∀x(x ∈ D → P(x))

2 “∃x ∈ D such that P(x)” can be written as

∃x(x ∈ D ∧ P(x))