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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 )....
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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.
- Spring '09