复旦大学：《离散数学》PPT教学课件（赵一鸣）05/28

Example: 1) Let 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, o Theorem3.3: Let g be a(everywhere)function from A to B, andf 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

Proof:(1) For every a∈A, If there exist x, y∈C such that(a,x)∈ fogang(a2y)∈∫g, then x=y? (2) For any a∈A, there exists c∈ C such that (a2c)∈f? o 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 fog is called a(everywhere) function firom a to o, we write fog:A→C.Ifa∈A, then(°g)(a)=f(g(a)

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