GO Classes Set Theory And Algebra Practice Set 1 | Question: 6

Question

Let $A$ be any set. Subset Relation on $\mathrm{P}(\mathrm{A})$ is Anti-symmetric?

Answer 1

Given a relation $R$ on a set $A, R$ is called antisymmetric if for all $a, b \in A,(a R b$ and $\mathrm{bRa}) \Rightarrow a=b$.

For a universe $U$, define the relation $R$ on $P(U)$ by $(A, B) \in R$ if $A \subseteq B$, for $(A, B) \subseteq U$

So $R$ is the subset relation if $A R B$ and $B R A$, then we have $A \subseteq B$ and $B \subseteq A$, which gives us $A=B$.

Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.