Equivalence relations on a set

Question

I am to find how many equivalence relations are there on a set $\left\{1, 2, 3\right\}$. How it can be counted? I will appreciate a step by step solution because I'm new to set theory. Thanks!

Answer 1

As commented befotre, every partition of a set $X$ defines one equivalence relation on $X$ and, conversly. The different partitions of $X=\{1,2,3\}$ are: $$\begin{aligned}&P_1=\left\{\{1,2,3\}\right\}\\ &P_2=\{ \{{2,3}\},\; \{1\}\}\\ &P_3=\{\{{1,3\}},\; \{{2\}}\}\\ &P_4=\{\{3\} ,\; \{1,2\}\}\\ &P_5=\{\{3\} ,\; \{2\},\; \{1\}\}. \end{aligned}$$