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HW5 - y 1,y 2 ∈ R assume that f-1 y 1 ≤ f-2 y 2 and...

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Math 23b Homework 5 Spring, 2009 Due Wednesday, March 18. Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/, w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the problem does not explicitly ask you to do so. 1. Problem 8 Page 95. Expand the formulas for f ◦ g and g ◦ f . 2. Problem 11 Page 95. Explain this in mathematical language in no more than 5 lines. 3. We say that a function g : R → R is increasing when the following holds: for x 1 , x 2 ∈ R , if x 1 > x 2 then g ( x 1 ) > g ( x 2 ). Let f : R → R be a function which is bijective and increasing. In no more than 20 lines, prove that the inverse function f - 1 : R → R is increasing. (Try to use a contrapositive argument.
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Unformatted text preview: y 1 ,y 2 ∈ R , assume that f-1 ( y 1 ) ≤ f-2 ( y 2 ), and then argue that y 1 ≤ y 2 .) 4. Let m,n ∈ N , and A , B be disjoint finite sets. Let f : [ m ] → A and g : [ n ] → B be bijections. In no more than 20 lines, give a bijection h : [ m + n ] → A ∪ B ; you must prove that the h you give, is a bijection. (Experiment with the case m = 2, n = 3.) 5. Let A and B be finite sets, whose cardinalities are m,n ∈ N respectively. In no more than 10 lines, prove that the cardinality of A ∪ B does not exceed m + n . (Note that A ∪ B = A ∪ ( B-A ); A and B-A are disjoint; B-A ⊂ B .) 6. Problem 1 Page 95....
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