第一題:
Let A ={1,2,3,4}and B={a,b,c,d,e}.Let f:A→B be defined by f={(1,d),(2,e),(3,d),(4,b)} and g:B→A {(a,2),(b,1),(c,3),(d,1),(e,4)}.List the ordered paris in f。g and g。f .

第二題:
Let R be the relation on Z*N defined by (a,b) R (c,d) if ad=bc,then prove that R is an equivalence relation.

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訪客 (未驗證)

2002-10-22 09:42
In First Problem:
The function f。g :B -->B and g。f : A -->A
thus, f。g={(a,e), (b,d), (c,d), (d,d), (e,b)}
Try g。f yourself!

The Second Proiblem:
Let R be the relation on Z*N defined by (a,b) R (c,d) if ad=bc,then prove that R is an equivalence relation.

Proof:
1. since aa=aa, => (a,a) R (a,a) for (a,a) in Z*N
2. If (a,b) R (c,d) then ad=bc => bc=ad, thus (a,b) R (c,d).
3. If (a,b) R (c,d), and (c,d) R (e,f), for all of them are in Z*N, then we have ad=bc and de=cf,
since af=ade/c = bce/c = be,
thus (a,b) R (e,f).

by 1.~3. we have prove that the Relation R is an
equi. relation on Z*N.