I'm determining whether the relation is: reflexive, symmetric, transitive or anti-symmetric.
Let $X$ be a non-empty set and let $\mathcal{P}(X)$ be the power set of $X$. Let $R_3$ be the relation defined on $\mathcal{P}(X)$ as follows: $\forall A, B\in \mathcal{P}(X)$, $(A, B) \in R_3$ if and only if $A \neq B$.
I've got that it is reflexive because $A R_3A $ since $A\neq B$ and $A=A$ therefore reflexive.
It's not anti-symmetric because $A$ and $B$ have to be distinct values, but I'm not sure how to prove for symmetric and transitive.
I was thinking since $A\neq B$ then $A>B$ or $A<B$.
Any help would be appreciated.
Best Answer
The relation is: