 2. Inverse functions  Inverse relation,  Function is a relation  Is the function’s inverse relation a function? No  Example: A={1,2,3},B={a,b}, f:A→B, f={(1,a),(2,b),(3,b)} is a function, but inverse relation f -1 ={(a,1),(b,2),(b,3)} is not a function.

 Theorem 3.7: Let f be a function from A to B, then inverse relation f -1 is a function if only if f is one-to-one correspondence.  Proof: (1)If f –1 is a function, then f is one-to-one correspondence.  (i)f is onto.  For any b  B ， there exists a  A such that f (a)=?b  (ii)f is one to one.  If there exist a 1,a 2  A such that f(a 1 )=f(a 2 )=b  B, then a 1 ?=a 2  (2)If f is one-to-one correspondence ， then f –1 is a function  f -1 is a function, for any b  B ， there exists one and only a  A so that (b,a)  f -1.  For any b  B, there exists a  A such that (b,a)  ?f -1.  For b  B ， If there exist a 1,a 2  A such that (b,a 1 )  f -1 and (b,a 2 )  f -1,then a 1 ?=a 2

 Definition 3.5: Let f be one-to-one correspondence between A and B. We say that inverse relation f -1 is the inverse function of f. We denoted f -1 ： B→A. And if f (a)=b then f -1 (b)=a.  Theorem 3.8: Let f be one-to-one correspondence between A and B. Then the inverse function f -1 is also one-to-one correspondence.  Proof: (1) f –1 is onto (f –1 is a function from B to A  For any a  A,there exists b  B such that f -1 (b)=a)  (2)f –1 is one to one  For any b 1,b 2  B, if b 1  b 2 then f -1 (b 1 )  f -1 (b 2 ).  If f:A→B is one-to-one correspondence, then f -1 ： B→A is also one-to-one correspondence. The function f is called invertible.

 Theorem 3.9: Let f be one-to-one correspondence between A and B.  Then  (1)(f -1 ) -1 = f  (2)f -1  f=I A  (3)f  f -1 =I B  Proof: (1)  (2)

 Let f:A→B and g:B→A ，  Is g the inverse function of f ?  f  g?=I B and g  f ?=I A  Theorem 3.10 ： Let g be one-to-one correspondence between A and B, and f be one-to-one correspondence between B and C. Then (f  g) -1 = g -1  f -1  Proof: By Theorem 3.6, f  g is one-to-one correspondence from A to C  Similarly, By theorem 3.7, g -1 is a function from B to A, and f –1 is a function from C to B.

 Theorem 3.11: Let A and B be two finite set with |A|=|B|, and let f be a function from A to B. Then  (1)If f is one to one, then f is onto.  (2) If f is onto, then f is one to one.  The prove are left your exercises.

 Exercise: P176 21,22  Prove T 3.11  Cardinality  Paradox  Pigeonhole principle P88 3.3