equivalence class on an equivalence relation?

Question

I have the relation $R = \{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}$ on the set $\{1,2,3,4\}$. I have to find [1] and [4] but I don't really understand what that means.

I get that $[a]$ is the set of all elements of $A$ related (by $R$) to $a$ so $[a]=\{x\in A : x$ $R$ $a\}$ right? But I don't get the significance of the number 1 or 4 inside of the brackets ([1] and [4]).

Answer 1

Your R relationship tells you which two elements are "equal" to each other, for instance 1=1, 1=2, 2=1, and so on...

[1] is a set of all the elements that are equal to 1 (that is the first element of all the pairs in R that have the second element 1), so [1] = {1, 2}. Also [4]={4}

Answer 2

$[1]$ is the set of all things that are equivalent to $1$, given the relation. So, since $1R1$ and $1R2$ are the only relations given that involve $1$, $[1] = \{1, 2\}$ (and in particular note that $[1] = [2]$). You can do the same thing for $[4]$.