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Unformatted text preview: MACM 101 Discrete Mathematics I Outline Solutions to Exercises on Propositional Logic 1. Construct a truth table for the following compound statement: ( p q ) ( q r ) . p q r ( p q ) ( q r ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2. Determine whether the following compound statement is a tautology ( p ( p q )) q. No, it is not. Method 1. Construct a truth table. Method 2. Use logical equivalences: ( p ( p q )) q ( p ( p q )) q expression for implications p q absorption law q p contrapositive. Now, if q = 1 and p = 0 , the implication is false. 3. Show that ( p r ) ( q r ) and ( p q ) r are logically equivalent. Method 1. Construct the truth tables for both statements and compare. Method 2. Use logical equivalences. ( p r ) ( q r ) ( p r ) ( q r ) expression for implications ( p q ) r idempotent law + associative law ( p q ) r De Morgans law ( p q ) r expression for implication 4. Show that ( p q ) r and ( p r ) ( q r ) are not logically equivalent. Do not use truth tables. Method 1. It is sufficient to find one assignment of values to p,q,r such that the two statements get different truth values. For instance if p = 0 , q = 1 , r = 0 then p q ) r = 1 while ( p r ) ( q r ) = 0 . Method 2. (Not recommended.) Use logical equivalences to simplify the biconditional (( p q ) r ) (( p r ) ( q r )) . It can be shown to be equivalent to ( p q ) ( p q ) r , that is, as is easily seen, not a tautology....
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This note was uploaded on 10/14/2011 for the course MACM 101 taught by Professor Pearce during the Spring '08 term at Simon Fraser.
- Spring '08