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x2solnfall2009

# x2solnfall2009 - ii B ⊆ A 3 i No f is not one-to-one...

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Concordia University Department of Computer Science & Software Engineering COMP 232/2 Mathematics for Computer Science FALL 2009 Examination 2 1. Prove that the sum of an integer and an irrational number must be irrational. 2. Let A and B be sets. i) Express { x | x A x / B } as simply as possible in the notation of set theory, without using set-builder notation. ii) Express x ( x A x / B ) as simply as possible in the notation of set theory, without any quantifiers. 3. Let f : R R be given by f ( x ) = 1 + x 2 . i) Is f one-to-one? Justify your answer. ii) Is f onto? Justify your answer. 4. Show for every real number x that x/ 2 = x / 2 . Hint: Consider separately the cases x odd and x even. Solutions 1. Assume n is an integer and u is irrational. If n + u were rational then u
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Unformatted text preview: ii) B ⊆ A 3. i) No, f is not one-to-one, since f (-1) = √ 2 = f (+1), but-1 6 = +1. ii) No, f is not onto, since f ( x ) = 0 = ⇒ 0 = 1 + x 2 = ⇒ x 2 =-1 = ⇒ F . 4. Let n = b x c and let r = x-n . It follows that x = n + r and 0 ≤ r < 1. If n is even then n = 2 k for some k ∈ Z and therefore ³ x 2 ´ = ³ n + r 2 ´ = ³ 2 k + r 2 ´ = ³ k + r 2 ´ = k = ³ 2 k 2 ´ = ³ n 2 ´ = ³ b x c 2 ´ since 0 ≤ r/ 2 < 1 / 2, and if n is odd then n = 2 k + 1 for some k ∈ Z and therefore ³ x 2 ´ = ³ n + r 2 ´ = ³ 2 k + 1 + r 2 ´ = ³ k + 1 + r 2 ´ = k = ³ 2 k + 1 2 ´ = ³ n 2 ´ = ³ b x c 2 ´ since 1 / 2 ≤ (1 + r ) / 2 < 1. 29-Oct-2009, 2:57 pm...
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