Let R$_1$ and R$_2$ be two equivalence relations on the same set A.
Not sure how to interpret this statement. Does it mean...
A = {1, 2} $\quad$#for example
AR$_1$A = {(1, 1), (1, 2), (2, 1), (2, 2)}
AR$_2$A = {(1, 1), (1, 2), (2, 1), (2, 2)}
R$_1$$\cap$R$_2$ = {(1, 1), (1, 2), (2, 1), (2, 2)}
R$_1$$\cup$R$_2$ = {(1, 1), (1, 2), (2, 1), (2, 2)}
When we say, for example, "let $R$ be an equivalence relation on the set $X$", we mean:
Suppose we have a subset $R\subseteq X\times X$ such that:
1) $\forall x\in X$ we have $(x,x)\in R$;
2) If $(x_1,x_2) \in R$, then $(x_2,x_1) \in R$;
3) If $(x_1,x_2),(x_2,x_3) \in R$, then $(x_1,x_3)\in R$.
So your specific statement is saying you have two such relations (or subsets, not necessarily the same) of a set $A$.