Given a non-empty set X. Consider P(X), which is the set of all subset of X. Defined the relation R in P(X) as follows:For subsets A and B in P(X),ARB if and only if A⊂B. Is R an equivalence relation on P(X)? Justify your answer.
Question
Given a non-empty set X. Consider P(X), which is the set of all subset of X. Defined the relation R in P(X) as follows:For subsets A and B in P(X),ARB if and only if A⊂B. Is R an equivalence relation on P(X)? Justify your answer.
Solution
An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity. Let's examine each of these properties for the relation R.
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Reflexivity: For any subset A in P(X), it is true that A⊂A. Therefore, the relation R is reflexive.
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Symmetry: For any subsets A and B in P(X), if A⊂B, it is not necessarily true that B⊂A. Therefore, the relation R is not symmetric.
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Transitivity: For any subsets A, B, and C in P(X), if A⊂B and B⊂C, then it is true that A⊂C. Therefore, the relation R is transitive.
Since the relation R is reflexive and transitive but not symmetric, it is not an equivalence relation on P(X).
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