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Unformatted text preview: MATH 290 HOMEWORK 8 DUE 22 APRIL 2010 Answer all of the following questions clearly and completely. You will be graded on both the answers themselves and on presentation of the answers. This includes style, grammar and spelling. There is to be no collaboration on this assignment. All work you hand in must be entirely your own. This assignment is due at the beginning of class on Thursday, April 22. No late assignments will be accepted. Let A and B be sets, and suppose that f : A B , and g : B C . 1. (a) Prove that if f and g are injective, then g f is injective. (b) Prove that if g f is injective, then f is injective. (c) Give an example of sets A , B , and C , and functions f : A B , and g : B C such that f is injective, g is not injective, but g f is injective. This shows that the result in part (b) is the best we can hope to get. (d) Give an example of sets A , B , and C , and functions f : A B , and g : B C such that f is injective, g is not injective, and g f is not injective. This shows that converse of the result in part (b) does not hold....
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