rcaXB Ris a relation from a to B DemRa。 (a,b)∈R(a,C)∈R (a2b)∈R(a,C)∈ Runless b=c ☆ function DomR=A,(everywhere)function
❖ RA×B,R is a relation from A to B， DomRA。 ❖ (a,b)R (a, c)R ❖ (a,b)R (a, c)R unless b=c ❖ function ❖ DomR=A， (everywhere)function
Chapter 3 Functions 今3.1 Introduction g Definition3. 1: Let A and b be nonempty sets. A relation is a(everywhere)function from A to B, denoted by f:A→>B, if for every a∈A, there is one and only b eB so that(a, bEf, we say that b-f(a). The set a is called the domain of the function f. If XCA, then f(X)--falaeX is called the image of X. The image of A itself is called the range of f, we write RelfYcB, thenf(Y=af(aEY, is called the preimage of Y A function f: A>B is called a mapping. If(a, b) fso that b=f(a), then we say that the element a is mapped to the element b
Chapter 3 Functions ❖ 3.1 Introduction ❖ Definition3.1: Let A and B be nonempty sets. A relation is a (everywhere)function from A to B, denoted by f : A→B, if for every aA, there is one and only b B so that (a,b) f, we say that b=f (a). The set A is called the domain of the function f. If XA, then f(X)={f(a)|aX} is called the image of X. The image of A itself is called the range of f, we write Rf . If YB, then f -1 (Y)={a|f(a)Y} is called the preimage of Y. A function f : A→B is called a mapping. If (a,b) f so that b= f (a), then we say that the element a is mapped to the element b
'(everywhere)function 令(1)D0m广=A; 令(2if(a,b)and(a,b)∈f, then b=b 令 Relation:(a,b),(a,b)∈R, 今 function:if(a,b)and(a,b)∈f, then b=b ☆ Relation: LorCa &(everywhere)function: DomR-A
❖ (everywhere)function: ❖ (1)Domf=A； ❖ (2)if (a,b) and (a,b')f, then b=b‘ ❖ Relation: (a,b),(a,b')R, ❖ function : if (a,b) and (a,b')f, then b=b‘ ❖ Relation: DomRA ❖ (everywhere)function: DomR=A
Example: Let A=1, 233, 49,B-a,b, cl R1=(1,a)(2,b),(3,c)}, R2={(1,a)2(1,b)(2,b),(3,c)2(4,c)}, R3={(1,a),(2b),(3,b)4,a)} Example: LetA=(-2, -1, 0, 1, 2) and B={0,1,2,3,4,5}. 令Let广={(-2,0),(-1,1),(0,0),(1,3),(2,5)}.fisa (everywhere)function 今X={-2,0,1},f(X)= Y={0,5},f1(Y)=?
❖Example：Let A={1,2,3,4},B={a,b,c}, ❖R1={(1,a),(2,b),(3,c)}, ❖R2={(1,a),(1,b),(2,b),(3,c),(4,c)}, ❖R3={(1,a),(2,b),(3,b),(4,a)} ❖Example: Let A ={-2,-1, 0,1,2} and B={0,1,2,3,4,5}. ❖ Let f={(-2,0),(-1,1), (0,0),(1,3),(2,5)}. f is a (everywhere)function. ❖ X={-2,0,1}, f(X)=? ❖ Y={0,5}, f -1 (Y)=?
&o Theorem 3. 1: Let f be a(everywhere) function from a to b and a and a be subsets of a. then %(1)IfAcA2, then f(ADcf(a2) (2)f(A1∩A2)∈f(A1)nf(A2) (3)f(A1UA2)=f(A1)∪f(A2) (4)f(A1)-f(A2)∈f(A1-A2) o Proof: 3)(a)f(AlUf(ACfA, UA2) 令(b)fA1UA2)∈fA1Uf(42)
❖Theorem 3.1: Let f be a (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∪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 )
(4)f(A1)-f(4A2)Cf(A1-A2 for any y∈∫(A1)f(A2)
❖ (4) f (A1 )- f (A2 ) f (A1 -A2 ) ❖ for any y f (A1 )-f (A2 )
Theorem 3.2: Let f be a(everywhere) function from a to B, andAcA(i-1, 2,,n). Then (1)f(4)=Uf(4) (2)f(4)∩f(4) i=1
❖ Theorem 3.2：Let f be a (everywhere) function from A to B, and AiA(i=1,2,…n). Then n i i n i f Ai f A 1 1 (1) ( ) ( ) = = = n i i n i f Ai f A 1 1 (2) ( ) ( ) = =
2. Special Types of functions .o Definition 3.2: Let a be an arbitrary nonempty set The identity function on A, denoted by Ia, is defined by la(a=a Definition 3.3. Let f be an everywhere function from A to B. Then we say that f is onto(surjective ifRFB. We say that f is one to one(injective) if we cannot have fad-fa2) for two distinct elements a and az ofA. Finally, we say that f is one-to-one correspondence(bijection), if f is onto and one-to one . &o The definition of one to one may be restated in the following equivalent form 今If(a1)=f(a2) then a1=a2 for all a,a2∈AOr 令Ifa1≠a2 then j(a1)≠f(a2) for all a,a2∈A
❖ 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-toone. ❖ 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), fa=ial; 令2)Letg:R( the set of real numbers→C(the set of complex number), ga=ia; 今3)Leth:Z一Zm={0,1,…m-1},h(a)= a mod n . s onto. one to one
❖ 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?
%o 3.2 Composite functions and Inverse functions %o1 Composite functions o Relation, Composition, g 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 og is a (everywhere)function from A to C
❖ 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