# Difficulty understanding notations for equivalence classes

So, this is the question I need to answer, and I am having trouble understanding the notations used, especially this line:

[a]i denotes the equivalence class of a under Ri(i = 1,2)

Given an equivalence relation (in terms of ordered pairs), I know how to find the equivalence classes. But I am unable to understand how to obtain the equivalence relation here, and what I am supposed to do after that.

Here is the entire question:

A is a set, and R1,R2 ⊆ A x A are equivalence relations on A. For a ∈ A, [a]i denotes the equivalence class of 'a' under Ri(i = 1,2) and [a] denotes the equivalence class of 'a' under R1 ∩ R2. Define [a] in terms of [a]1 and [a]2

• Welcome to MSE. If you need help formatting math on this site, here's a tutorial Commented Oct 24, 2020 at 23:31
• The question can be rephrased as follows. Let $[a]_1$ be the equivalence class of $a$ with respect to $R_1$ and $[a]_2$ the equivalence class of $a$ with respect to $R_2$. $R_1\cap R_2$ is also an equivalence relation; let $[a]$ be the equivalence class of $a$ with respect to $R_1\cap R_2$. Express $[a]$ in terms of the sets $[a]_1$ and $[a]_2$. Does this help at all? Commented Oct 24, 2020 at 23:55

## 1 Answer

I guess since you are new to this topic, maybe the intuition isn't there yet. Here is a simple example that you can try, and hopefully it helps you get the answer.

Let $$A$$ be the set of integers. Let $$R_1$$ be the equivalence relation of 'congruent modulo 2'. We denote the equivalence classes by $$[n]_2$$. What is the set represented by $$[0]_2$$?

$$[0]_2 =$$ the even numbers $$= \{ -2, 0, 2, 4, 6, \dots \}$$.

Let $$R_2$$ be the equivalence relation of 'congruent modulo 3'. We denote the equivalence classes by $$[n]_3$$. What is the set represented by $$[0]_3$$?

$$[0]_3 =$$ the multiples of $$3$$ $$= \{ -3, 0, 3, 6, \dots \}$$.

What is the equivalence relation defined by $$R_3 = R_1 \cap R_2$$?

It is the equivalence relation of 'congruent modulo 6'.

Denote this equivalence relation by $$[n]_6$$. What is the set represented by $$[0]_6$$?

$$[0]_6 =$$ the multiples of $$6$$ $$= \{ -6, 0, 6, 12, \dots \}$$.

Now, what is the relation between $$[0]_2$$, $$[0]_3$$, and $$[0]_6$$?

$$[0]_6 = [0]_2 \cap [0]_3$$

From this answer, you can guess the answer to your original question. Now try to turn it into a proof!