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Homework_set_1_typeset.pdf_

Homework_set_1_typeset.pdf_ - f[0 1 →(0 1 Problem 2 Show...

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Math 413, Fall semester, 2007 Bigtimes Homework set 1, handed out August 23. due August 30 Read all of Chapter 1, and the first 2 sections of Chapter 2. You should find it Chapter 1 easy going, but 2.1 ande especially 2.2 are rather more demanding. In the text, do exercise 3 of section 1.1.3. This is a bit tricky, especially (d) if you haven’t seen how to do it. In Section 1.2.3, do exercises 1 (you might use part (c) above, even if you didn’t manage to prove it), 2, 4, 5. Problem 1. Find an explicit bijective map
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Unformatted text preview: f : [0 , 1] → (0 , 1). Problem 2. (*) Show that if X and Y are two sets, and both f : X → Y and g : Y → X are injective, then there exists a bijective map h : X → Y . This is really quite tricky; think of going from X to Y to X . . . and forming chains as long as you can. How do these chains begin and end? To how many chains can an element of X or Y belong? Finally, in Section 2.1.3, do exercises 1, 3, 4, 6, 7. These are easy as compared to the previous ones....
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