Answer to Question #146316 in Discrete Mathematics for Promise Omiponle

Question #146316
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.
1
Expert's answer
2020-11-29T19:24:08-0500

Let S={a,b,c,d,e}S =\{a, b, c, d, e\} , and P={{a,b},{c,d},{e}}.P=\{\{a, b\},\{c, d\},\{e\}\}.


(a) By defenition, a partition of a set SS is a set of non-empty subsets of SS  such that every element xx  in SS is in exactly one of these subsets. Since {a,b},{c,d},{e}\{a, b\},\{c, d\},\{e\} are non-empty set and each element sSs\in S is in exactly one of the sets {a,b},{c,d}\{a, b\},\{c, d\} and {e}\{e\}, PP really is a partiton of SS.


(b) Let us find the equivalence relation RR on SS induced by PP. By defenition, (x,y)R(x,y)\in R if and only if xx and yy 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)}.R=\{(a,a),(a,b),(b,a),(b,b), (c,c),(c,d),(d,c),(d,d),(e,e)\}.



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