Solution to Let S ={a, b, c, d, e}, and P={{a, b},{c, d},{e}}. (a) Verify that P … - Sikademy
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Archangel Macsika

Let S ={a, b, c, d, e}, and P={{a, b},{c, d},{e}}. (a) Verify that P really is a partiton of S. (b) Find the equivalence relation R on S induced by P.

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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\in 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)\in 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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