# HW 1 - MATH 290 HOMEWORK 1 SOLUTIONS 1 Show the following...

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MATH 290 – HOMEWORK 1 – SOLUTIONS 1. Show the following using a truth table. (a) ( P Q ) = R is equivalent to ( P = R ) ( Q = R ) (b) P = ( Q R ) is equivalent to ( P ∧ ∼ Q ) = R ). Solution: (a). P Q R P Q P = R Q = R P Q = R ( P = R ) ( Q = R ) T T T T T T T T T T F T F F F F T F T F T T T T T F F F F T T T F T T F T T T T F T F F T F T T F F T F T T T T F F F F T T T T (b). P Q R Q R P ∧ ∼ Q P = ( Q R ) P ∧ ∼ Q = R T T T T F T T T T F T F T T T F T T T T T T F F F T F F F T T T F T T F T F T F T T F F T T F T T F F F F F T T 2. Consider the following sentence taken from Form 1040, “ If you file a complete and accurate tax return and you are due a refund, your refund will be issued within 40 days if you file a paper return or within 21 days if you file electronically. (a) Write the above statement as a propositional form clearly defining simple, relevant propositions by the letters P , Q , R , etc. For example, you might say “Let P = ‘you filed a complete tax return’, and Q = ‘you filed an accurate tax return.’ ” etc. (b) Write a useful denial, in propositional form, of the result of part (a).

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