I am studying Naive Set Theory and stuck to understand algebraic part of equivalence relation. Can you please explain what these inclusions mean?
The algebra of relations provides some amusing formulas. Suppose that, temporarily, we consider relations in one set $X$ only, and, in particular, let $I$ be the relation of equality in $X$ (which is the same as the identity mapping on X). The relation I acts as a multiplicative unit; this means that $IR = RI = R$ for every relation $R$ in $X$. Query: is there a connection among $I$, $RR^{–1}$, and $R^{–1}R$? The three defining properties of an equivalence relation can be formulated in algebraic terms as follows: reflexivity means $I ⊂ R$, symmetry means $R ⊂ R^{–1}$, and transitivity means $RR ⊂ R$.
Halmos, Paul R.. Naive Set Theory (Dover Books on Mathematics) (p. 41). Dover Publications. Kindle Edition.
In set theory a binary relation on a set $X$ is some (any) subset of $X\times X.$ We can write $I_X=\{(x,x):x\in X\}$, which is a subset of $X\times X.$ For subsets $R, S$ of $X\times X$ we can write $$R^{-1}=\{(y,x): (x,y)\in R\}$$ and $$RS=\{(x,z):\exists y\,(\,(x,y)\in R\land (y,z)\in S\,)\}.$$ In particular $RR=\{(x,z):\exists y\,(\,(x,y)\in R\land (y,z)\in R\}$.
We can also show that a binary relation $R$ on $X$ is an equivalence relation iff $I_X\subseteq R=R^{-1}=RR.$
BTW. In topology $I_X$ is called the diagonal of $X\times X.$