On the power set P(A) of a set A,the relation | vedclass

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(B) Let $R$ be the relation defined on $P(A)$ such that $X R Y$ if and only if $X \subseteq Y$ for all $X, Y \in P(A)$.$1$. Reflexivity: For any $X \i

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Anti-symmetric

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(B) Let $R$ be the relation defined on $P(A)$ such that $X R Y$ if and only if $X \subseteq Y$ for all $X, Y \in P(A)$.$1$. Reflexivity: For any $X \in P(A)$,$X \subseteq X$ is always true. Thus,$R$ is reflexive.$2$. Anti-symmetry: If $X \subseteq Y$ and $Y \subseteq X$,then by the definition of set equality,$X = Y$. Thus,$R$ is anti-symmetric.$3$. Transitivity: If $X \subseteq Y$ and $Y \subseteq Z$,then $X \subseteq Z$. Thus,$R$ is transitive.Since the relation is reflexive,anti-symmetric,and transitive,it is a partial order relation,not an equivalence relation. Therefore,the relation is anti-symmetric.

On the power set $P(A)$ of a set $A$,the relation "is a subset of" $(\subseteq)$ is:

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