Let $S$ be the equivalence relation defined on $\wp(\{1, 2, 3, 4\})$ defined by: $$XSY\text{ if and only if } |X|\equiv|Y|\;\mod 2$$ Write down the equivalence classes of S.
I understand that equivalence classes have relations where it is reflexive, symmetric and transitive but how are you supposed to write equivalence classes?
On
In particular, there are only two equivalence classes of $S$. One is, denoted by $\overline{0}$, $$\{ \emptyset, \{1, 2\}, \{ 1, 3\}, \{1, 4\}, \{2,3\},\{2,4\},\{3,4\},\{1,2,3,4\} \}.$$ And the other one is, denoted by $\overline{1}$ $$ \{ \{1\},\{2\},\{3\},\{4\}, \{2,3,4\},\{1,3,4\},\{1,2,4\},\{1,2,3\}\}.$$
What you're calling the "absolute value of a set" is actually referred to as its cardinality. For finite sets, which is the relevant case here, cardinality is how many elements are in the set. It gets a bit murkier with infinite sets but I won't bog you down with the details here.
So in short: you look at the power set of $\{1,2,3,4\}$, i.e. the set of subsets of that. You then define the relation $S$ on these sets, where two sets, $X,Y$ are related if their cardinalities satisfy $|X| \equiv |Y| \pmod 2$.
Equivalently, let $X$ have $n$ elements and $Y$ have $m$ elements, where $X,Y \in P(\{1,...,4\})$. Then $XSY$ if and only if $n \equiv m \pmod 2$.