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 )