Let $\mathscr{P} \left( A \right)$ denote the set of all subsets of a set $A$.
Show that the relation $R$ defined on $\mathscr{P} \left( A \right)$ by $a R b, \forall x (x \in A \Leftrightarrow x \in B)$ is an equivalence relation on $\mathscr{P} \left( A \right)$.
My idea is to let $a \in \mathscr{P} (A), b \in \mathscr{P} (A)$. Then $a R a$; $a R b \Leftrightarrow b R a$; $a R b, b R c \Leftrightarrow a R c$. But how to organize mathematical language？Can you give me some advice? Thank you.
(1) Reflexivity. Let subset $B \subseteq A$ be arbitrary; we want to show $B R B$. This is trivial: letting $x \in A$ be arbitrary, if $x \in B$, then obviously $x \in B$.
(2) Symmetry. Let subsets $B, C \subseteq A$ be such that $B R C$; we want to show $C R B$. Again, this is trivial: if $x \in C$, then by hypothesis, $x \in B$. Conversely, if $x \in B$, then by hypothesis, $x \in C$. Since for all $x$, we have that $(x \in C) \Leftrightarrow (x \in B)$, then $C R B$.
(3) Transitivity. Let subsets $B, C, D \subseteq A$ be such that $B R C$ and $C R D$; we want to show that $B R D$. Let $x \in B$. Then because $B R C$, we have $x \in C$; and because $C R D$, we have that $x \in D$. Conversely, let $x \in D$. Because $C R D$, we have $x \in C$; and because $B R C$, we have $x \in B$. So for all $x$, we have that $(x \in B) \Leftrightarrow (x \in D)$, showing that $B R D$.
This kind of equivalence is just the familiar notion of "set equivalence": roughly speaking, two sets are equivalent if they contain the same elements.