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Given a non-empty set X, consider P(X) w...

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 you answer

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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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