# Finding the equivalence classes of the relation R

Find the equivalence classes of the relation R = {(0, 0),(1, 1),(1, 2),(2, 2),(2, 1),(3, 3),(3, 4),(4, 3),(4, 4)}

on the set A = {0, 1, 2, 3, 4}.

How do i solve this question. I'm attempting to teach myself at the moment so any help will be appreciated.

As far as i'm aware the equivalence class of a is the set of all elements x in A, such that x is related to a by r.

• just like you said, which are the elements that $1$ is related to, for example? May 20, 2019 at 2:45
• For a equivalence relation, the relation itself is reflexive ($aRa$ is true), symmetric ($aRb \implies bRa$), and transitive ($aRb, bRc \implies aRc$). All at once. May 20, 2019 at 3:10
• Since (1,1) is in R, 1 is related to itself (and in general x is related to itself, for every equivalence relation). For the given R, 1 is also related to 2 (and 2 is related to 1). 1 is related to no other elements (apart from 1 and 2). So the equivalence class of 1 is {1,2}. Since 0 is only related to itself, its equivalence class is {0}. All equivalence classes are {0}, {1,2} and {3,4}. Now let me give you a different problem, what are the equivalence classes for the relation Q={(5,6),(6,5),(7,7),(8,8),(5,5),(6,6)}? May 20, 2019 at 13:07

## 2 Answers

The equivalence class of $$x$$, denoted $$[x]$$, is the set of all elements of $$A$$ that are related to $$x$$. More formally, $$[x] = \{y \in A | (x,y) \in R\}$$.

Looking at $$R$$, we see that $$1$$ is related to $$2$$ and $$3$$ is related to $$4$$, so we can ‘combine’ the equivalence classes for $$1$$ and $$2$$, and for $$3$$ and $$4$$. We can ignore all of the other pairs as they are simply the result of the fact that $$R$$ is an equivalence relation—they don’t give us any more information. We have $$[1] = [2]$$ and $$[3] = [4]$$ and so our equivalence classes are

$$[0] = \{0\}$$ $$[1] = \{1,2\}$$ $$[2] = \{1,2\}$$ $$[3] = \{3,4\}$$ $$[4] = \{3,4\}$$

An equivalence relation on $$A$$ induces a partition on $$A$$, so we may also show the equivalence classes by writing

$$\{\{0\},\{1,2\},\{3,4\}\}$$

As you said in the question, we form equivalence classes by finding elements in $$R$$ related to different elements of $$A$$.

So from the relation $$R$$ we find that $$(0,0) \in R \Rightarrow 0 \in [0].$$ Also, $$(1,2) \in R \Rightarrow 2 \in [1]\ \text{and}\ 1 \in [2]$$ etc. Therefore, $$[0] = \{0\}, [1] = \{1,2\}$$ etc.

Here $$[\cdot]$$ denotes an equivalence class and '$$\cdot$$' is the representative of the class.

Can you find other classes like $$[2]$$ and $$[3]$$ from here?

• An equivalence class of an element of $A$ should be a subset of $A$, rather than a set of pairs in $R$. So $[0] = \{0\}$ rather than $\{(0,0)\}$, for example. May 20, 2019 at 3:47
• I just made a blunder, I would correct it now, thanks.
– Rick
May 20, 2019 at 3:50