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Unformatted text preview: that g ◦ f = 1 A . Prove that f is one-to-one. (b) Suppose that f : A-→ B and g : B-→ A are functions such that f ◦ g = 1 B . Prove that f is an onto function. 7. Let f : A-→ B be an onto function. Prove that there is a function g : B-→ A such that f ◦ g = 1 B . 8. Suppose that f : A-→ B and g : B-→ C are both bijections. Prove that g ◦ f : A-→ C is also a bijection. 9. Suppose that A has at most two elements, and f,g : A-→ A are bijections. Prove that f ◦ g = g ◦ f . 10. Suppose that A has 3 elements. Construct bijections f,g : A-→ A such that f ◦ g = g ◦ f . 11. Prove that if A and B are both countable, then so is A × B . 12. Prove that for any set A , there is no bijection between A and P ( A ). Also prove that there is always an one to one function from A to P ( A ). 2...
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This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.
- Spring '10