sol-midterm1 - a from the first set with a vertex b from...

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MACM 101 — Discrete Mathematics I Outline Solutions to Midterm 1 1. How to prove that a statement x P ( x ) where P ( x ) is a predicate is false? We have to prove that P ( a ) is false for every value a from the universe. 2. Simplify ( ¬ p ( p q )) q . Use logical equivalences: ( ¬ p ( p q )) q ⇐⇒ (( ¬ p p ) ( ¬ p q )) q distributive law ⇐⇒ ( ¬ p q ) q law of contradiction and domination law ⇐⇒ ¬ ( ¬ p q ) q expression for implication ⇐⇒ ( p ∨¬ q ) q DeMorgan’s law and double negation law ⇐⇒ T associative law, law of excluded middle and domination law 3. Draw Venn diagram of the symmetric difference of sets A and B . See the slides. 4. State DeMorgan’s laws of set theory. A B = A B A B = A B 5. Show that ( p q ) ( r s ) and ( p r ) ( q s ) are not logically equivalent. If p = 0 , q = 1 , r = 0 , s = 0 , then ( p q ) ( r s ) = 1 , while ( p r ) ( q s ) = 0 . 6. What is the graph of a binary relation? The graph of a binary relation R from A to B consists of two sets of vertices labeled by the elements of the set A and the set B , respectively, an edges or arcs connecting a vertex
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Unformatted text preview: a from the first set with a vertex b from the second set if and only if ( a, b ) ∈ R . 7. What relevant conclusion can be drawn from the following premises: “I am either dreaming or hallucinating.” “If I am dreaming, I am snoring.” “If I am hallucinating, I see elephants running down the road.” “I am not snoring.” The relevant conclusion is: “I see elephants running down the road.” Let the primitive statements be: d , ‘I am dreaming’ h , ‘I am hallucinating’ s , ‘I am snoring’ e , ‘I see elephants running down the road’ Then the premises are translated as: d ∨ h , d → s , h → e , ¬ s . And the conclusion: e . Steps Reason 1. d → s premise 2. ¬ s premise 3. ¬ d modus tollens to Steps 1 and 2 1 4. d ∨ h premise 5. h rule of disjunctive syllogism 6. h → e premise 7. e modus ponens to Steps 5 and 6. 2...
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This note was uploaded on 10/09/2009 for the course MATH macm 101 taught by Professor Jcliu during the Spring '09 term at Simon Fraser.

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sol-midterm1 - a from the first set with a vertex b from...

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