soln2 - MATH 135 Algebra Solutions to Assignment 2 1(a Make...

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Unformatted text preview: MATH 135 Algebra, Solutions to Assignment 2 1: (a) Make a truth table for the statement ( P ↔ ¬ R ) ∧ ( R → Q ). Solution: Here is a truth table. P Q R ¬ R P ↔ ¬ R R → Q ( P ↔ ¬ R ) ∧ ( R → Q ) T T T F F T F T T F T T T T T F T F F F F T F F T T T T F T T F T T T F T F T F T F F F T F T F F F F F T F T F (b) Determine whether ( P ∧ ¬ Q ) ∨ ( R ↔ P ) is equivalent to Q ↔ R . Solution: We make a truth table for ( P ∧ ¬ Q ) ∨ ( R ↔ P ) and Q ↔ R . P Q R ¬ Q P ∧ ¬ Q R ↔ P ( P ∧ ¬ Q ) ∨ ( R ↔ P ) Q ↔ R T T T F F T T T T T F F F F F F T F T T T T T F T F F T T F T T F T T F F F F T F T F F F T T F F F T T F F F F F F F T F T T T Since the final two columns are not identical, the two statements are not equivalent (for example, as seen on the third row, when P is true, Q is false and R is true, the statement ( P ∧ ¬ Q ) ∨ ( R ↔ P ) is true but the statement Q ↔ R is false). (c) Determine whether P → ( Q → R ) is equivalent to ( P → Q ) → R . Solution: We make a truth table for the two given statements. P Q R Q → R P → ( Q → R ) P → Q ( P → Q ) → R T T T T T T T T T F F F T F T F T T T F T T F F T T F T F T T T T T T F T F F T T F F F T T T T T F F F T T T F Since the column for P → ( Q → R ) is not identical to the column for ( P → Q ) → R , these two statements are not equivalent (for example, as seen on the 6 th row, when P is false, Q is true and R is false, the statement P → ( Q → R ) is true but the statement ( P → Q )...
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soln2 - MATH 135 Algebra Solutions to Assignment 2 1(a Make...

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