Friday, 29 April 2016

functions - Partition of Set: Proof




Let f:AB. If {B1,B2,,Bn} is a partition of B, prove that {f1(B1),f1(B2),,f1(Bn)} is partition of A.





I approached it like following:
Since f1 exists f must be one one and onto. So for each x in A there is one distinct image in B under A. Converse, "for each y in B there is distinct pre-image under f". This implies that we can define a set AiA which is the set of all elements in A whose image lies in Bi under the f i.e., Ai=1f(Bi). Since f is onto f covers all of B and hence union of Ais is equal to A.
Since Bi is non-empty set, there exist a y in Bi such that f(x)=y. This means that x is pre-image of y under f. Hence x belongs to Ai. So Ai is non empty set.
Now since BiBj is empty if ij, them there is no y such that it belongs to both Bi and Bj. Now f is one one so there is no pre-image x such that it belongs to both Ai and Aj. So AiAj is empty. Hence set of all Ais form a partition of A.



I think my logic is correct but can someone help me writing it in formal way. Also since there is one one correspondence between equivalence relations and partitions, how to approach this using relations.


Answer



No, here f is not necessarily a bijection. This is my hint: show that if X,YB then
f1(XY)=f1(X)f1(Y)andf1(XY)=f1(X)f1(Y)
where the preimage f1(X):={aA:f(a)X}.
Can you take it from here?




Edit. As regards the intersection-property, see how to prove f1(B1B2)=f1(B1)f1(B2) . The union-property can be shown in a similar way.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...