1 / 13

R  A×B,R is a relation from A to B ， DomR  A 。 (a,b)  R (a, c)  R

R  A×B,R is a relation from A to B ， DomR  A 。 (a,b)  R (a, c)  R (a,b)  R (a, c)  R unless b=c function DomR = A ， (everywhere) function 。. Chapter 3 Functions. 3.1 Introduction

derrj
Télécharger la présentation

R  A×B,R is a relation from A to B ， DomR  A 。 (a,b)  R (a, c)  R

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. RA×B,R is a relation from A to B，DomRA。 • (a,b)R (a, c)R • (a,b)R (a, c)R unless b=c • function • DomR=A，(everywhere)function。

  2. 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 aA, 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 XA, then f(X)={f(a)|aX} is called the image of X. The image of A itself is called the range of f, we write Rf. If YB, 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.

  3. (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: DomRA • (everywhere)function: DomR=A

  4. 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)=?

  5. Theorem 3.1: Let f be a (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)

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

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

  8. 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

  9. 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?

  10. 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.

  11. Proof: (1)For any aA, there exists cC such that (a,c) f g? • (2)For every aA, If there exist x,yC such that (a,x)f gand (a,y)f g，then x=y? • 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)).

  12. 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

  13. Exercise: P176 2,9,10,13,14, • 28,37,38 • Next: Inverse functions • The Characteristic function of the set P178 5.2 • Cardinality • Paradox

More Related