Logic and Discrete Mathematics. Section 3.5 Propositional logical equivalence Negation of propositional formulae
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1 Logic and Discrete Mathematics Section 3.5 Propositional logical equivalence Negation of propositional formulae Slides version: January 2015
2 Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A B, if they obtain the same truth value under any truth valuation (of the variables occurring in them).
3 Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A B, if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: (p q) p q
4 Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A B, if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: (p q) p q p q (p q) p q T T F T T T F T F F T T F T T F F F T T T F F T T F F T T F T F T F F T F F F T F T T F
5 Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A B, if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: (p q) p q p q (p q) p q T T F T T T F T F F T T F T T F F F T T T F F T T F F T T F T F T F F T F F F T F T T F p (p q) p p p
6 Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A B, if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: (p q) p q p q (p q) p q T T F T T T F T F F T T F T T F F F T T T F F T T F F T T F T F T F F T F F F T F T T F p (p q) p p p p q p (p q) p p T T T T T T T T T T T F T T T T F T T T F T F F F T T F F F F F F F F F F F F F
7 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A
8 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A Logical equivalence is reducible to logical validity: A B iff = A B.
9 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A Logical equivalence is reducible to logical validity: A B iff = A B. is an equivalence relation, i.e., reflexive, symmetric, and transitive.
10 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A Logical equivalence is reducible to logical validity: A B iff = A B. is an equivalence relation, i.e., reflexive, symmetric, and transitive. Moreover, is a congruence with respect to the propositional connectives, i.e.:
11 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A Logical equivalence is reducible to logical validity: A B iff = A B. is an equivalence relation, i.e., reflexive, symmetric, and transitive. Moreover, is a congruence with respect to the propositional connectives, i.e.: if A B then A B, and
12 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A Logical equivalence is reducible to logical validity: A B iff = A B. is an equivalence relation, i.e., reflexive, symmetric, and transitive. Moreover, is a congruence with respect to the propositional connectives, i.e.: if A B then A B, and if A 1 B 1 and A 2 B 2 then (A 1 A 2 ) (B 1 B 2 ), where {,,, }.
13 Some basic properties of logical equivalence Logical equivalence is reducible to logical consequence: A B iff A = B and B = A Logical equivalence is reducible to logical validity: A B iff = A B. is an equivalence relation, i.e., reflexive, symmetric, and transitive. Moreover, is a congruence with respect to the propositional connectives, i.e.: if A B then A B, and if A 1 B 1 and A 2 B 2 then (A 1 A 2 ) (B 1 B 2 ), where {,,, }. Theorem for equivalent replacement: Let A, B, C be any propositional formulae p be a propositional variable. If A B then C(A/p) C(B/p), where C(X/p) is the result of simultaneous substitution of al occurrences of p by X.
14 Some important logical equivalences Idempotency: p p p; p p p.
15 Some important logical equivalences Idempotency: p p p; p p p. Commutativity: p q q p; p q q p.
16 Some important logical equivalences Idempotency: Commutativity: Associativity: p p p; p p p. p q q p; p q q p. (p (q r)) ((p q) r); (p (q r)) ((p q) r). Note that this property allows us to omit the parentheses in multiple conjunctions and disjunctions.
17 Some important logical equivalences Idempotency: Commutativity: Associativity: p p p; p p p. p q q p; p q q p. (p (q r)) ((p q) r); (p (q r)) ((p q) r). Note that this property allows us to omit the parentheses in multiple conjunctions and disjunctions. Absorption: p (p q) p; p (p q) p.
18 Some important logical equivalences Idempotency: Commutativity: Associativity: p p p; p p p. p q q p; p q q p. (p (q r)) ((p q) r); (p (q r)) ((p q) r). Note that this property allows us to omit the parentheses in multiple conjunctions and disjunctions. Absorption: Distributivity: p (p q) p; p (p q) p. p (q r) (p q) (p r); p (q r) (p q) (p r).
19 Other useful logical equivalences A A ; A A A A; A A ; A A A A A B (A B) (B A) A B A B A B A B A B ( A B); A B ( A B) A B B A
20 Important equivalences for negations of propositional formulae A A, (A B) A B, (A B) A B, (A B) A B, (A B) ((A B) (B A)) (A B) (B A) (A B) (B A).
21 Important equivalences for negations of propositional formulae A A, (A B) A B, (A B) A B, (A B) A B, (A B) ((A B) (B A)) (A B) (B A) (A B) (B A). Using these equivalences, all occurrences of negations in any propositional formula can be driven inwards, so eventually they only occur in from of propositional variables. Then the formula is transformed to a negation normal form.
22 Transformation to negation normal form: example ( p ( q r))
23 Transformation to negation normal form: example ( p ( q r)) p ( q r) p ( q r) p (q r)
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