Let $R=\{(1,1),(2,2),(3,3),(4,4),(2,3),(3,2),(2,4),(4,2),(3,4),(4,3)\}$ be an equivalence relation on $A=\{1,2,3,4\}$

Question

Let $R=\{(1,1),(2,2),(3,3),(4,4),(2,3),(3,2),(2,4),(4,2),(3,4),(4,3)\}$ be an equivalence relation on $A=\{1,2,3,4\}$. How many elements does $[2]$ have?

My answer: By drawing a graph I can see that $2R2$, $2R3$ and $2R4$. So $[2]=\{2,3,4\}$, which means $|[2]|=3$.

Q1: Am I wrong?

Q2: Other solution methods?

Q3: Also, I'm not sure whether $2 \in [2]$ or not. It should be because $2R2$, since it is an equivalence relation.

Answer 1

Your solution is correct and the class of $[2]$ has three elements as you have suggested.

Over all you may find the partition of $$\{1,2,3,4\}$$ by this relation as $$\{ \{1\}, \{2,3,4\}\}$$

Thus you only have two equivalency classes.

Answer 2

Q1: No, You are right.

Q2: No real other solution. Others, in fact, use same concepts.

Q3: Since Reflexivity is a part of equivalency, as you see $(2,2) \in R$, so $x \in [x]$ always.