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Question
Given a non-empty set X, consider P(X), which is the set 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 your answer.
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Solution
(i) Reflexive:
Since every set is a subset of itself, ARA for all A ∈ P(X).
∴ R is reflexive.
(ii) Symmetric:
Let ARB ⇒ A ⊂ B
This cannot be implied to B ⊂ A.
For instance, if A = {1, 2} and B = {1, 2, 3}, then it cannot be implied that B is related to A.
= ARB ≠ BRA
∴ R is not symmetric.
(iii) Transitive:
Further, if ARB and BRC, then A ⊂ B and B ⊂ C.
⇒ A ⊂ C
⇒ ARC
∴ R is transitive.
Hence, R is not an equivalence relation to P(X).
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