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Given a non-empty set X, consider P(X) which is theset of all subsets of X. Define the relation R in P(X) as follows:For subsets A, B in P(X), ARB if and only if A B. Is R an equivalence relation on P(X)?Justify you answer |
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Answer» Since every set is a subset of itself, ARA for all ` A in P(X).` Thererfore, R is reflexive. Let ` ARB implies A subset B.` This cannot be implied to `B subset A`. For instant, if `A={1,2}` and `B={1,2,3},` then it cannot be implied that B is related to A. Therefore, R is not symmetric. Further if ARB and BRC, then ` A subset B` and `B subset C`. `implies A subset C` `implies ARC` Therefore, R is transitive. Hence, R is not an equivalence relation since it is not symmetric. |
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