Question

Given a non-empty set X. Consider P(X), which is the set of all subset of X. Defined the relation R in P(X) as follows:
For subsets A and B in P(X),ARB if and only if $A\subset B$. Is R an equivalence relation on P(X)? Justify your answer.

Answer 1

Since, every set is a subset of itself, ARA for all $A\in P\left(X\right).$
Therefore, R is reflexive.

$ARB\Rightarrow A\subset B$
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 $B\subset C.$
$\Rightarrow A\subset C\Rightarrow ARC$. Therefore, R is transitive.
Hence, R is not an equivalence relation since, it is not symmetric.