Hi,

May I get some help from the following algebra homework? Thanks.

1 EXERCISE 1: EQUIVALENCE RELATIONS ALWAYS COME FROM MAPS 1.1 PROBLEM Let S be a set. Recall that if T is a further set, and if f : S } T is a map, th- E denotes the relation 1' on 3 dened by (ash) 4: (were). This is an equivalence relation, called \"equality upon applying f\" or \"equality under f\". Now, let m be any equivalence relation on S. Prove that N has the form for a properly chosen set T and a properly chosen 3" : S } T. More precisely, prove that N equals E, where T is the quotient set 3;\" N and where f fzSi-Tistheprojectionrnapa'w : SrSfm. [Hint: To prove that two relations R1 and R2 on S are equal, you need to check that every pair (a, b) of elements of 5' satises the equivalence (anb) 4:)- (agb