# Defining relations on a power set

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.

• What does $A!$ mean for an arbitrary set $A$? – DanielWainfleet Oct 8 '15 at 7:53
• @user254665 I believe the author uses a pseudo-code convention and means $A \neq B$. – GenericNickname Oct 8 '15 at 7:55
• @user254665 $A != B$ is how you often write $A \neq B$ in programming languages. – mrp Oct 8 '15 at 7:55
• Yes that is correct. – user3491814 Oct 8 '15 at 7:58

• not reflexive, since $(A,A) \in R_3 \Rightarrow A \neq A$, which obviously is not true.
• symmetric, since $(A,B) \in R_3 \Rightarrow A \neq B \Rightarrow B \neq A \Rightarrow (B,A) \in R_3$
• not anti-symmetric: Since $X \neq \emptyset$, we know that $X,\emptyset \in P(X)$ and $(X,\emptyset), (\emptyset, X) \in R_3$ but (as stated before) $X \neq \emptyset$.
• not transitive: $(A,B), (B,C) \in R_3 \Rightarrow A\neq B$ and $B \neq C$ but this does not imply $A \neq C$. For example take $A = C = \emptyset$ and $B = X$. Then $(A,B),(B,C) \in R_3$ but $(A,C) \notin R_3$.