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,...

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

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1 EXERCISE 1: EQUIVALENCE RELATIONS ALWAYS COMEFROM 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 defined 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 ffzS—i-Tistheprojectionrnapa'w : S—rSfm.[Hint: To prove that two relations R1 and R2 on S are equal, you need to check thatevery pair (a, b) of elements of 5' satisfies the equivalence (anb) 4:)- (aflgbfl