Question:
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:
Solution:
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 A = {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.