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2. Inverse functions Inverse relation, Function is a relation

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.

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2. Inverse functions Inverse relation, Function is a relation

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  1. 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.

  2. 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 a1,a2A such that f(a1)=f(a2)=bB, then a1?=a2 • (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 a1,a2A such that (b,a1) f-1 and (b,a2) f-1,then a1?=a2

  3. 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 aA,there exists bB such that f -1(b)=a) • (2)f –1 is one to one • For any b1,b2B, if b1b2 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.

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

  5. 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 (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.

  6. 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.

  7. Exercise: P176 21,22 • Prove T 3.11 • Cardinality • Paradox • Pigeonhole principle P88 3.3

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