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# 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.

Q.8 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 $A \subset B$. Is R an equivalence relation

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Given a non empty set X, consider P(X) which is the set of all subsets of X.

Since, every set is subset of itself , ARA  for all $A \in P(x)$

$\therefore$  R is reflexive.

Let $ARB \Rightarrow A\subset B$

This is not same as $B\subset A$

If  $A =\left \{ 0,1 \right \}$      and    $B =\left \{ 0,1,2 \right \}$

then we cannot say that B is related to A.

$\therefore$ R is not symmetric.

If $ARB \, \, \, and \, \, \, BRC, \, \, then \, \, A\subset B \, \, \, and \, \, B\subset C$

this implies $A\subset C$  $= ARC$

$\therefore$ R is transitive.

Thus, R is not an equivalence relation because it is not symmetric.

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