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Exam s 1-4

# Exam s 1-4 - (d State in English the contrapositive of the...

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MATH 290 – 20 MAY 2010 – EXAM 1 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This test will be counted out of a total of 60 points. 1. (8 pts.) Prove that P = ( Q R ) is equivalent to ( P ∧ ∼ Q ) = R by ﬁlling in the truth table below. P Q R Q R P = ( Q R ) P ∧ ∼ Q ( P ∧ ∼ Q ) = R T T T T T F T F T T F F F T T F T F F F T F F F 2. (8 pts. each) Consider the statement: If a + b is even, then both a and b are even or both a and b are odd. (a) Translate the above statement into symbolic form. (b) Identify the antecedent and consequent of the statement. (c) State in English the converse of the statement.
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Unformatted text preview: (d) State in English the contrapositive of the statement. 3. (8 pts. each) Consider the following sentence: For every natural number n , there is a non-negative integer k and an odd integer m such that n = 2 k m . (a) Translate the above sentence into symbolic form. (b) Write a useful denial of the sentence in symbolic form and translate it into English. 4. (8 pts.) Show that [ Q ∧ ( P = ⇒ Q )] = ⇒ P is neither a tautology nor a contradiction....
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