s5 - Selected solutions to Homework 5 1#12 in Appendix B...

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Selected solutions to Homework # 5 1. #12 in Appendix B. Compute the image of each of the following functions: (a) f : R R ; f ( x ) = x 2 . The image is the set of non-negative real numbers. (b) f : Z Q ; f ( x ) = x - 1. The image is the set of integers (as a subset of Q : im f = { q Q : q is an integer } . (c) f : R R ; f ( x ) = - x 2 + 1. The image is the set of real numbers less than or equal to 1: im f = { x R : x 1 } . 2. #14 in Appendix B. (a) Give an example of a function which is injective, but not surjective. An example would be f : R R ; f ( x ) = tan - 1 x . (b) Give an example of a function which is surjective, but not injective. An example would be f : R R ; f ( x ) = x 3 - x . 3. #2 in Section 1.3 Let p Z , p 6 = - 1 , 0 , 1. Prove that p is prime if and only if for each a Z either ( a, p ) = 1 or p | a . Proof: Suppose that p is prime. Let a be an integer which is not divisible by a . Since the only divisors of p are - 1 , 1 , - p, p , the only common divisors of a are - 1 and 1. Hence ( a, p ) = 1. Since this argument works for every integer which does not divisible by p , we
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