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
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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
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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”
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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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