Let S={a,b,c,d,e} , and P={{a,b},{c,d},{e}}.
(a) By defenition, a partition of a set S is a set of non-empty subsets of S such that every element x in S is in exactly one of these subsets. Since {a,b},{c,d},{e} are non-empty set and each element s∈S is in exactly one of the sets {a,b},{c,d} and {e}, P really is a partiton of S.
(b) Let us find the equivalence relation R on S induced by P. By defenition, (x,y)∈R if and only if x and y are elements of the same set of a partition. In our case, R={(a,a),(a,b),(b,a),(b,b),(c,c),(c,d),(d,c),(d,d),(e,e)}.
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