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# lect3 - DeMorgans Laws p q(p(q(p q(p(q p q pq 1 1 1 1 0 0...

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1 DeMorgan’s Laws ¬ ( p q ) ( ¬ p ) ( ¬ q ) ¬ ( p q ) ( ¬ p ) ( ¬ q ) p q p q ¬ ( p q ) ¬ p ¬ q ( ¬ p ) ( ¬ q ) 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 ¬ (Peter is tall and fat) Peter is not tall Peter is not fat ¬ (cucumbers are green or seedy) cucumbers are not green cucumbers are not seedy

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2 Other important logical equivalences p q ¬ ( p ¬ q ) (proof by contradiction) p q p q ¬ p ¬ p q 1 1 1 0 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 ¬ p q
3 What is negation of implication? ¬ ( ¬ p q ) p ( ¬ q ) ¬ ( p q )

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4 p q q p converse of p q p q ¬ p ¬ q inverse of p q p q p q q p 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 ¬ p ¬ q ¬ p ¬ q 0 0 1 0 1 1 1 0 0 1 1 1 An integer is divisible by 4 it is divisible by 2. An integer is divisible by 2 it is divisible by 4. An integer is not divisible by 4 it is not divisible by 2.
5 p q ( ¬ q ) ( ¬ p ) contrapositive of p q p q p q ¬ q ¬ p ( ¬ q ) ( ¬ p ) 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 Contrapositive law An integer is no divisible by 2 it is not divisible by 4.

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6 Double negation: ¬¬ p p p q = ¬ p ¬ q = ¬ ¬ p 1 0 1 0 1 0
7 Commutativity p q q p p q q p Associativity p ( q r ) ( p q ) r p ( q r

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lect3 - DeMorgans Laws p q(p(q(p q(p(q p q pq 1 1 1 1 0 0...

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