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# Hmk4 - COT 3100 Homework 4 Spring 2000 Assigned Due 3/30,31...

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COT 3100 Homework # 4 Spring 2000 Assigned: 3/7/00 Due: 3/30,31 in recitation 1) Define a relation T N x N such that T = {(a,b)| a A b A a – b = 2c+1 for some integer c}. (N is the set of non-negative integers.) a) Prove that this relation is not reflexive. b) Prove that this relation is symmetric. c) Define the term anti-transitive as the following: Given a set A and a relation R, if for all a,b,c A, (aRb bRc cRa) (a = b b = c) Prove that the relation T is anti-transitive. 2) Let g: A A be a bijection. For n 2, define g n = g ° g ° ... ° g, where g is composed with itself n times. Prove that for n 2, that g n is a bijection from A to A as well, and show that (g n ) -1 = (g -1 ) n . 3) Let f : A B and g: B C denote two functions. If both f and g are injective, prove that the composition g ° f: A C is an injection as well. 4)

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Hmk4 - COT 3100 Homework 4 Spring 2000 Assigned Due 3/30,31...

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