chapter 3 functions
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Chapter 3 Functions. P181(Sixth Edition) P168(Fifth Edition). Theorem 3.1: Let f be an everywhere function from A to B, and A 1 and A 2 be subsets of A. Then (1)If A 1 A 2 , then f (A 1 ) f (A 2 ) (2) f (A 1 ∩A 2 ) f (A 1 )∩ f (A 2 ) - PowerPoint PPT PresentationTRANSCRIPT
Chapter 3 Functions
P181(Sixth Edition) P168(Fifth Edition)
Theorem 3.1: Let f be an everywhere function from A to B, and A1 and A2 be subsets of A. Then
(1)If A1A2, then f(A1) f(A2)(2) f(A1∩A2) f(A1)∩f(A2)(3) f(A1 A∪ 2)= f(A1)∪f(A2)(4) f(A1)- f(A2) f(A1-A2) Proof: (3)(a) f(A1)∪f (A2) f(A1 A∪ 2) (b) f(A1 A∪ 2) f(A1)∪f (A2)
(4) f (A1)- f (A2) f (A1-A2) for any y f (A1)-f (A2)
Theorem 3.2 : Let f be an everywhere function from A to B, and AiA(i=1,2,…n). Then
n
ii
n
ii AfAf
11)()()1(
n
ii
n
ii AfAf
11)()()2(
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, a2A Or If a1a2 then f(a1)f(a2) for all a1, a2A
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?
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.
Proof: (1)For every aA, If there exist x,yC such that (a,x) f gand (a,y) f g , then x=y?
(2)For any aA, there exists cC 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 aA, then(f g)(a)=f(g(a)).
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(gf )=(hg)f
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 aA, (f IA)(a) ?
=f(a). Property (ii) is proved similarly to property
(i).
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 cC, there exists aA such
that f g(a)=c
(2)one to one:if ab , 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.
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: :(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,a2A such that f(a1)=f(a2)=bB, then
a1?=a2 (2)If f is one to one , then f –1 is a function f -1 is a function For bB , If there exist a1,a2A such that (b,a1)f -1
and (b,a2) f -1,then a1?=a2
Proof: (b)(1)If f –1 is an everywhere 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 an
everywhere function f -1 is an everywhere 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 (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 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 -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
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, 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.
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)
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 AU. The characteristic function of A is defined as a function from U to {0,1} by the following:
AxAx
xA 01
)(
Theorem 3.12: Let A and B be subsets of the universal set. Then, for any xU, 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 AB; (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);
);(1)()7( xx AA
)()()()()8( xxxx BAABABA
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, ParadoxPigeonhole principle P100(Sixth)P88
3.3(Fifth)