This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: that g ◦ f = 1 A . Prove that f is onetoone. (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...
View
Full
Document
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
 DrChanSongHeng

Click to edit the document details