# what is the absolute value of a set?

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?

• It is not the absolute value. It is the cardinality, i.e. (for finite sets) the number of its elements. Apr 30, 2019 at 7:57
• Presumably $|X|$ means the number of elements of $X$. If $X$ is in the power set of $A$ then its elements are some (or all or none) of the elements of $A$ Apr 30, 2019 at 7:58

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 $$n \equiv m \pmod 2$$.
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 $$\overline{1}$$ $$\{ \{1\},\{2\},\{3\},\{4\}, \{2,3,4\},\{1,3,4\},\{1,2,4\},\{1,2,3\}\}.$$
• You can see @Eevee Trainer's answer, $\overline{0}$ contains all the subsets of cardinalities 0,2 and 4, which are $0 \pmod{2}$. and $\overline{1}$ contains all the subsets of cardinalities 1 and 3, which are $1 \pmod{2}$. That's why I use the two notations, $\overline{0}$ and $\overline{1}$. Apr 30, 2019 at 8:58