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Fundamental Theorems of Functions: Properties, Composite and Inverse Relations

This section elaborates on critical theorems related to functions, including properties of image sets under functions, composite functions, and the definitions of onto, one-to-one, and one-to-one correspondence functions. Theorems provide necessary proofs for the relationships among subsets and how composite functions behave under various mappings. The role of inverse functions is addressed, emphasizing their conditions for being well-defined based on the injectiveness and surjectiveness of the original function. Examples illustrate key concepts for clarity.

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Fundamental Theorems of Functions: Properties, Composite and Inverse Relations

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  1. Chapter 3 Functions P181(Sixth Edition) P168(Fifth Edition)

  2. Theorem 3.1: Let f be an everywhere function from A to B, and A1 and A2 be subsets of A. Then • (1)If A1A2, then f(A1) f(A2) • (2) f(A1∩A2) f(A1)∩f(A2) • (3) f(A1∪A2)= f(A1)∪f(A2) • (4) f(A1)- f(A2) f(A1-A2) • Proof: (3)(a) f(A1)∪f (A2) f(A1∪A2) • (b) f(A1∪A2) f(A1)∪f (A2)

  3. (4) f (A1)- f (A2) f (A1-A2) • for any y f (A1)-f (A2)

  4. Theorem 3.2：Let f be an everywhere function from A to B, and AiA(i=1,2,…n). Then

  5. 2. Special Types of functions • Definition 3.2：Let A be an arbitrary nonempty set. The identity function on A, denoted by IA, is defined by IA(a)=a. • Definition 3.3.: Let f be an everywhere function from A to B. Then we say that f is onto(surjective) if Rf=B. We say that f is one to one(injective) if we cannot have f(a1)=f(a2) for two distinct elements a1 and a2 of A. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-to-one. • The definition of one to one may be restated in the following equivalent form: • If f(a1)=f(a2) then a1=a2 for all a1, a2A Or • If a1a2 then f(a1)f(a2) for all a1, a2A

  6. Example:1) Let f: R(the set of real numbers)→C(the set of complex number), f(a)=i|a|; • 2)Let g: R(the set of real numbers)→C(the set of complex number), g(a)=ia; • 3)Let h:Z→Zm={0,1,…m-1}, h(a)=a mod m • onto ,one to one?

  7. 3.2 Composite functions and Inverse functions • 1.Composite functions • Relation ,Composition, • Theorem3.3: Let g be a (everywhere)function from A to B, and f be a (everywhere)function from B to C. Then composite relation f g is a (everywhere)function from A to C.

  8. Proof: (1)For every aA, If there exist x,yC such that (a,x)f gand (a,y)f g，then x=y? • (2)For any aA, there exists cC such that (a,c) f g ? • Definition 3.4: Let g be a (everywhere) function from A to B, and f be a (everywhere) function from B to C. Then composite relation f g is called a (everywhere) function from A to C, we write f g:A→C. If aA, then(f g)(a)=f(g(a)).

  9. Since composition of relations has been shown to be associative (Theorem 2.), we have as a special case the following theorem. Theorem 3.4: Let f be a (everywhere) function from A to B, and g be a (everywhere) function from B to C, and h be a (everywhere) function from C to D. Then h(gf )=(hg)f

  10. Theorem 3.5: Let f be an everywhere function from A to B. Then (i)f IA=f. (ii)IB f = f. Proof. Concerning(i), let aA, (f IA)(a) ?=f(a). Property (ii) is proved similarly to property (i).

  11. Theorem 3.6: Let g be an everywhere function from A to B, and f be an everywhere function from B to C. Then (1)if f and g are onto , then f g is also onto. (2)If f and g are one to one, then f g is also one to one. (3)If f and g are one-to-one correspondence, then f g is also one-to-one correspondence Proof: (1)for every cC, there exists aA such that f g(a)=c

  12. (2)one to one:if ab，then f g(a)? f g(b) (3)f and g are one-to-one correspondence, f and g are onto. By (1), f g are onto. By (2),  f g are also one to one. Thus f g is also one-to-one correspondence.

  13. 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.

  14. Theorem 3.7: :(a) Let f be a function from A to B, Inverse relation f -1 is a function from B to A if only if f is one to one (b) Let f be an everywhere function from A to B, Inverse relation f -1 is an everywhere function from B to A if only if f is one-to-one correspondence. Proof: (a)(1)If f –1 is a function, then 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，then f –1 is a function f -1 is a function For bB，If there exist a1,a2A such that (b,a1)f -1 and (b,a2) f -1,then a1?=a2

  15. Proof: (b)(1)If f –1 is an everywhere 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 an everywhere function f -1 is an everywhere 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

  16. Definition 3.5: Let f be one-to-one (correspondence) between A and B. We say that inverse relation f -1 is the (everywhere) 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 an everywhere 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.

  17. 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)(f -1)-1= f • (2)f -1 f=IA • Let f:A→B and g:B→A， • Is g the inverse function of f ? • f g?=IB and g  f ?=IA

  18. 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, 3.8, g-1 is a one-to-one correspondence function from B to A, and f –1 is a one-to-one correspondence function from C to B.

  19. Theorem 3.11: Let A and B be two finite set with |A|=|B|, and let f be an everywhere 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(p189.36)

  20. 3.3 The Characteristic function of the set • function from universal set to {0,1} • Definition 3.6: Let U be the universal set, and let AU. The characteristic function of A is defined as a function from U to {0,1} by the following:

  21. Theorem 3.12: Let A and B be subsets of the universal set. Then, for any xU, we have • (1)A(x)0 if only if A= • (2)A(x)  1 if only if A=U; • (3)A(x)≦B(x) if only if AB; • (4)A(x) B(x) if only if A=B; • (5)A∩B(x)=A(x)B(x); • (6)A∪B(x)=A(x)+B(x)-A∩B(x);

  22. Exercise:P188 9,10,13,14, 21,22,28, 32,33,36,37,38(Sixth) • OR P176 9,10,13,14, 21,22,28, 32,33,36,37,38(Fifth) • Next: Cardinality, Paradox • Pigeonhole principle P100(Sixth)P88 3.3(Fifth)

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