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2. Let f: Z-Z be given by the following rule: x+6 x=0 x = 1 x+8 x 2 f(x)=x+5 (mod 3) (mod 3) (mod 3) For example, to find f(100), we determine first that 100 = 3.33+1 and therefore 100 = 1 (mod 3); therefore f(100) = 105. (a) Give an example to show that f is not injective. (b) Give an example to show that f is not surjective. (c) Change one of the numbers 6, 5, and 8 to turn f into a bijection.

2. Let f: Z-Z be given by the following rule: x+6 x=0 x = 1 x+8 x 2 f(x)=x+5 (mod 3) (mod 3) (mod 3) For example, to find f(100), we determine first that 100 = 3.33+1 and therefore 100 = 1 (mod 3); therefore f(100) = 105. (a) Give an example to show that f is not injective. (b) Give an example to show that f is not surjective. (c) Change one of the numbers 6, 5, and 8 to turn f into a bijection.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 30E: Let be as described in the proof of Theorem. Give a specific example of a positive element of .
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2. Let f: Z-Z be given by the following rule: x+6 x=0 f(x)=x+5 x = 1 x = 2 x+8 (mod 3) (mod 3) (mod 3) For example, to find f(100), we determine first that 100 = 3.33+1 and therefore 100 = 1 (mod 3); therefore f(100) = 105. (a) Give an example to show that f is not injective. (b) Give an example to show that f is not surjective. (c) Change one of the numbers 6, 5, and 8 to turn f into a bijection.
Transcribed Image Text:2. Let f: Z-Z be given by the following rule: x+6 x=0 f(x)=x+5 x = 1 x = 2 x+8 (mod 3) (mod 3) (mod 3) For example, to find f(100), we determine first that 100 = 3.33+1 and therefore 100 = 1 (mod 3); therefore f(100) = 105. (a) Give an example to show that f is not injective. (b) Give an example to show that f is not surjective. (c) Change one of the numbers 6, 5, and 8 to turn f into a bijection.
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