Equivalence class/Equivalence relation question.

Question

Let R be an equivalence relation on set A, and let a,b$\in$A. If b$\in$[a] then [b]$\subseteq$[a].

I need help proving this. Thank you in advance.

Answer 1

Hint:

By definition $[a]=\{b~:~b\simeq a\}$, in other words, it is the set of all things related to $a$.

So, suppose that $b\in [a]$, that means that $b\simeq a$.

Now... to prove that one set is a subset of another, take an arbitrary element in the one on the left and show that it must also be one of the right. Remember again that $[b]=\{c~:~c\simeq b\}$ and use your properties of equivalence relations.