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 bB ， there exists aA such that f (a)=?b (ii)f is one to one. If there exist a1,a2A such that f(a1)=f(a2)=bB, then a1?=a2

(2)If f is one-to-one correspondence ， then f –1 is a function f -1 is a function, for any bB ， there exists one and only

aA so that (b,a) f-1. For any bB, there exists aA such that (b,a)?f-1. For bB ， If there exist a1,a2A such that (b,a1) f-1 and

(b,a2) f-1,then a1?=a2

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 –1is onto (f –1 is a function from B to A For any aA,there exists bB such that f -1(b)=a) (2)f –1 is one to one For any b1,b2B, if b1b2 then f -1(b1) f -1(b2). 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 -1f=IA

(3)f f -1=IB

Proof: (1) (2)

Let f:A→B and g:B→A， Is g the inverse function of f ? f g?=IB and g f ?=IA 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 (fg)-1= g-1

f -1

Proof: By Theorem 3.6, fg 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 21,22Prove T 3.11

CardinalityParadoxPigeonhole principle P88 3.3