1090-11s-test1

# 1090-11s-test1 - Math 1090 A Summer 2011 Answers to Test 1...

This preview shows pages 1–3. Sign up to view the full content.

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: Math 1090 A Summer 2011 Answers to Test 1 Each question is worth 10 marks. 1. • Calculate (a) using the formal definition of substitution. Show each step. • For each of (b)–(d): if it’s defined, just write the result; if it’s not defined, explain why. (a) [4 marks] ¬ ( p ∧ ¬ q )[ p := r ] Answer ¬ ( p ∧ ¬ q )[ p := r ] is ¬ ( p [ p := r ] ∧ ( ¬ q )[ p := r ] ) (using the recursive step of the definition), which is ¬ ( p [ p := r ] ∧ ¬ q [ p := r ] ) (using the recursive step), which is ¬ ( p [ p := r ] ∧ ¬ q ) (using the basis step), which is ¬ ( r ∧ ¬ q ) (using the basis step). (After the first step, there are other possible orders of steps.) (b) [2 marks] ( p ≡ q ) ∨ q [ q := q → r ] Answer ( p ≡ q ) ∨ ( q → r ) (When doing the substitution, we need to put back the outer brackets of q → r , since ∨ has higher priority than → .) (c) [2 marks] p ≡ q ∨ q [ q := q → r ] Answer p ≡ q ∨ ( q → r ) (d) [2 marks] ( p ∧ q )[ p ∧ q := p ∧ q ] Answer Substitution (of a formula) for a formula which is not a variable (in a formula) is not defined. (Only substitution of a formula for a variable in a formula is defined.) 1 2. Use a truth table, or an argument based on truth tables, to determine whether, for all formulas A and B , each of the following is a tautology. (a) [5 marks] ( A → B ) ∨ ( B → A ) Answer 1 A B ( A → B ) ∨ ( B → A ) f f t t t f t t t f t f f t t t t t t t Column 4 shows that, for all formulas A and B , and for every state v , it’s true that v satisfies ( A → B ) ∨ ( B → A )....
View Full Document

## This note was uploaded on 07/31/2011 for the course MATH 1090 taught by Professor Georgetourlakis during the Spring '09 term at York University.

### Page1 / 6

1090-11s-test1 - Math 1090 A Summer 2011 Answers to Test 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online