sol-midterm1

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online