Give 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 $A$ and $B$ in $P (X)$ , $A R B$ if $A \subset B$ , is $R$ an equivalence relation on $P (X)$ ?
Justify your answer.

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Solution

Since $A \subset A \forall A \in P (X),$
$∴ A R A$ , $\forall A \in P (X)$
So, $R$ is reflexive.
Also for $A, B, C \in P (X),$ ARB and BRC
$\Rightarrow A \subset B$ and $B \subset C$
$\Rightarrow A \subset C \Rightarrow A R C$
$∴ R$ is transitive
But, $A R B$ does not imply $B R A$ .
Hence, $R$ is not an equivalence relation on $P (X)$ .

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Give 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 and B in P(X), ARB if A⊂B, is R an equivalence relation on P(X)? Justify your answer.

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Give 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:
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