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CSS342: Propositions3 Propositions Examples: –In math 1.The only positive integers that divide 7 are 1 and 7 itself: (true) 2.For every positive integer n, there is a prime number larger than n: (true) –In history 1.Alfred Hitchcock won an Academy Award in 1940 for directing “Rebecca”: (false, he has never won one for directing) 2.Seattle held the World’s Fair, Expo 62: (true ) –In programming languages 1.Boolean expressions in if-else, while, and for statements for ( index = 0; index < 100; index++ ) { …….; } A proposition Not a proposition
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CSS342: Propositions 1
CSS342: Propositions
Professor: Munehiro Fukuda
CSS342: Propositions 2
Introduction• Propositional logic is useful in CS to analyze such behavior
of code portions.• Proposition: a statement that is either true or false, but not
both• Logic: reasoning whether a consequence of given statements
is correct, but not whether each statement is true.• Example:
– Proposition p:All mathematicians wear sandals. (A → B)– Proposition q:Anyone who wears sandals is an algebraist (B → C)– From p and q:All mathematicians are algebraists. (A → C)
CSS342: Propositions 3
Propositions• Examples:
– In math1. The only positive integers that divide 7 are 1 and 7 itself: (true)2. For every positive integer n, there is a prime number larger than n:
(true)– In history
1. Alfred Hitchcock won an Academy Award in 1940 for directing “Rebecca”: (false, he has never won one for directing)
2. Seattle held the World’s Fair, Expo 62: (true )– In programming languages
1. Boolean expressions in if-else, while, and for statementsfor ( index = 0; index < 100; index++ ) {
…….;}
A proposition
Not a proposition
CSS342: Propositions 4
Compound propositions• Conjunction of p and q
– notations: p ^ q, p && q– True only if both p and q are true– truth table
• Disjunction of p or q– Notations: p v q, p || q– True if either p or q or both are true– truth table
p q p ^ q
F F F
F T F
T F F
T T T
p q p v q
F F F
F T T
T F T
T T T
CSS342: Propositions 5
Binary Expressions in C++ Part1• How do you examine the behavior of if-else?
– if ( a > = 1 && a <= 100 )• true if 1 ≤ a ≤ 100
– if ( a >= 1 || a <= 100 )• always true
a < 1 1 <= a <= 100 100 < a
a >= 1
a <= 100
false
true
true
true
true
false
conditionproposition
a < 1 1 <= a <= 100 100 < a
a >= 1
a <= 100
false
true
true
true
true
false
conditionproposition
CSS342: Propositions 6
Negation
• The negation of p– Notations: p, not p, ¬p !p– Truth table:
• Example:– P: 1 + 1 = 3 (false)– !p: !(1 + 1 = 3) ≡ 1 + 1 ≠ 3 (true)
P !p
F T
T F
CSS342: Propositions 7
Boolean Algebra (Axioms)
When T = true and F = false:1. If p ≠ F then p = T, If p ≠ T then p = F2. F &&F = F, T || T = T3. T && T = T, F || F = F4. F && T = T && F = F5. F || T = T || F = T6. !T = F, !F = T
CSS342: Propositions 8
Boolean Algebra (Theorems)When T = true and F = false:• p && q = q && p, p || q = q || p• (p && q) && r = p && (q && r), (p || q ) || r = p || (q || r)• (p || q) && (p || r) = p || q && r, p && q || p && r = p && (q ||
r)• p && F = F, p || T = T• p && T = p, p || F = p• p && !p = F, p || !p = T• p && p = p, p || p = p• p && (p || q) = p, p || p && q = p• !(!p) = p• p + q && !p = p + q
Commutative Law
Associative Law
Distributive Law
Complement Law
Double Negation
Absorption Law
Square Law
Consensus Law
CSS342: Propositions 9
Binary Expressions in C++ Part2• How do you examine the behavior of if-else
statements?– Given two distance objects, d1 and 2, if ( d1 < d2 )
p: d1.feet < d2.feetq: d1.feet == d2.feeta: d1.inches < d2.inchesb: d1.inches == d2.inchesc: d1.inches > d2.inches
false
false
false
false
false
true
true
true
true
d1 < d2d2.inchesd1.inchesd2.feetd1.feet
<<
<
<
<
<=
==
=
==>>>
>
>
>
CSS342: Propositions 10
Binary Expressions in C++ Part2(Cont’d)
bool Distance::operator < ( const Distance& d2 ) const {return ( feet < d2.feet ) || (feet == d2.feet) && (inches < d2.inches);
}
p: d1.feet < d2.feetq: d1.feet == d2.feeta: d1.inches < d2.inchesb: d1.inches == d2.inchesc: d1.inches > d2.inches
p && a || p && b || p && c || q && a Distributive Law= p && (a || b || c ) || q && a !a = b || c= p && (a || !a ) || q && a Complement Law= p && T || q && a Theorem 6= p || q && a= ( d1.feet < d2.feet ) || ( d1.feet == d2.feet ) && (d1.inches < d2.inches)
CSS342: Propositions 11
Conditional Propositions• Given two propositions such as p and q,
If p then q or p → q
is called a conditional proposition.– P: hypothesis, antecedent, or sufficient condition– Q: conclusion, consequent, or necessary condition
• if p then q ≡ p only if q: called logically equivalent.– John may take CSS342 only if he advances to CSS343.– If John takes CSS342, he advances to CSS343.
qp
CSS342: Propositions 12
Truth Table of Conditional Propositions
• Chair statement: if CSS gets an additional $80,000, it will hire one new faculty member.
• P: CSS gets an additional $80,000.• Q: CSS will hire one new faculty member.• If both p and q are true, the chair said a correct statement.• If p is true but q is false, the chair said a wrong statement.• If p is false, the chair is not responsible for his statement. We
should regard it as true. p q p → qT T TT F FF T TF F T
CSS342: Propositions 13
• Two propositions: P: 1 > 2 Q: 4 < 8
• According to the truth table, p → q is true, while q → p is false.
• C++ program:• What’s the execution result?
– Linux:• 1 > 2 -> 4 < 8: 8• 4 < 8 -> 1 < 2: 0
– Goodall• 1 > 2 -> 4 < 8: 0• 4 < 8 -> 1 < 2: 0
• It depends on how result1 wasinitialized.
• In math, we consider all initialized in true.
Binary Expressions in C++ Part3
void main( ) { bool result1; if ( 1 > 2 ) result1 = 4 < 8; cout << "1 > 2 -> 4 < 8: " << result1 << endl;
bool result2; if ( 4 < 8 ) result2 = 1 > 2; cout << "4 < 8 -> 1 > 2: " << result2 << endl;}
CSS342: Propositions 14
Logical Equivalence• If two different compound propositions have the same
truth values no matter what truth values their constituent propositions have, they are called: – logically equivalent
• Example:– !(p || q) ≡ !p && !q (De Morgan’s Laws)
p q p || q !(p || q) !p !q !p && !q
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T T T
CSS342: Propositions 15
De Morgan’s Laws
• !(p || q) ≡ !p && !q– Proved by the truth page on the previous page.
• !(p && q) ≡ !p || !q
p q p && q !(p && q) !p !q !p || !q
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
CSS342: Propositions 16
Biconditional Propositions• Given two propositions such as p and q,
p if and only if q p iff q or p ↔ qis called a conditional proposition.
• P: a necessary and sufficient condition for Q• Q: a necessary and sufficient condition for P
p q p ↔ qT T TT F FF T FF F T
CSS342: Propositions 17
Biconditional Propositions Proof of the Truth Table
p q p → q q → p (p → q) && (q → p) p ↔ q
T T T T T T
T F F T F F
F T T F F F
F F T T T T
CSS342: Propositions 18
Contrapositive
• !q →!p– The contrapositive of p → q– Logically equivalent
p q p → q !q !p !q →!pT T T F F TT F F T F FF T T F T TF F T T T T
