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


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 AB in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify you answer:


Since every set is a subset of itself, $A R A$ for all $A \in \mathrm{P}(X)$.

∴R is reflexive.

Let $A R B \Rightarrow A \subset B$.

This cannot be implied to $B \subset A$.

For instance, if = {1, 2} and B = {1, 2, 3}, then it cannot be implied that B is related to A.

∴ R is not symmetric.

Further, if $A R B$ and $B R C$, then $A \subset B$ and $B \subset C$.

$\Rightarrow A \subset C$

$\Rightarrow A R C$

∴ R is transitive.

Hence, R is not an equivalence relation since it is not symmetric.

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