Chapter 3 Functions 令P181( Sixth edition &P168 (Fifth Edition)
Chapter 3 Functions ❖ P181(Sixth Edition) ❖ P168(Fifth Edition)
Theorem 3.1: Let f be an everywhere function from a to b and a and a be subsets of a. then &(1)IfA,cA2, then f(ACf(A,2) (2)f(A1A2)f(A1)f(A2) (3)f(A1UA2)=f(A1)∪f(A2) (4)f(A1)-f(A2)∈f(A1-A2) o Proof:(3)(afADUf(AcA, UA2) 令(b)fA1UA2)∈fA1Uf(42)
❖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∪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 今 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 an everywhere function from A to B, and A cA(i-1, 2,o.n). Then (1)f(4)=Uf(4) (2)f(4)∩f(4) i=1
❖ Theorem 3.2：Let f be an 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 a, 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 fa-f(az) for two distinct elements a and a2 of A. 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 a1,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