# L2[1] - Example Show that p p q and p q are logically...

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Example Show that and are logically equivalent by developing a series of logical equivalences. ( ( )) p p q pq ( ( ( )) (De Morgan) p p q p p q [ ( ) ] (De Morgan) p p q [ (double negation) p p q ( ) ( ) (distributive) p p p q ( ) (negation) F p q ( ) (commutative) p q F (identity)

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More Useful Equivalences Involving conditionals Involving biconditionals p q p q (implication contrapositive) p q q p ( ) ( ) p q p q q p p q p q
Example Show that is a tautology. p q p q ( ) ( ) (previous slide) p q p q p q p q ( ) ( ) (De Morgan) p q p q ( ) (associative) p q p q ( ) (commutative) p p q q ( ) ( ) (associative x2) p p q q (negation x2) TT (domination) T

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Example Prove Absorption Law (second one) using other known laws. That is, show that () p p q p ( ) ( ) ( ) (identity) p p q p F p q (domination,commutative) pF (Identity) p ( ) (distributive) p F q
Example Show that and are not logically equivalent. Solution: all we need to do is find a set of truth values for p, q, r, s that lead to different truth values for the two compound propositions given. (i.e. a counterexample). If r is true and p, q, s are false, then the first statement is false and the second statement is true. ( ) ( ) p q r s ( ) ( ) p r q s

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Predicates Some statements involve variables x>3 x=y+3 x+y=z “computer x is under attack by an intruder” “computer x is functioning properly”
Predicates Looking at the first example, which can be written in plain English as “x is greater than 3”, we see that it can be broken into two parts.

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L2[1] - Example Show that p p q and p q are logically...

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