What is a simple example that shows equivalence classes constitute a partition?

Question

Can someone illustrate using a simple concrete example that the equivalence classes defined by $\sim$ constitute a partition of a set $A$?

Answer 1

Consider the set $A = \{a,b,c,d,e,f,g,h,i \}$, and look at the array: $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}$$Define $\sim$ on $A$ by saying that $x \sim y \iff x$ and $y$ are in the same column.

This way, we have $a \sim d$, $b \sim h$, $f \sim i$, $a \not\sim e$, $h \not\sim c$, for example. I'll leave you to check that $\sim$ is an equivalence relation as an exercise. Denote by $[x]$ the set $\{y \in A \mid x \sim y \}$. This way we have: $$[a] = [d] = [g] = \{a,d,g\},$$and similarly for the other two columns equivalence classes (can you guess what they are?). We can partition $A$ as $[a]\cup [e]\cup [f]$, or $[d]\cup [b] \cup [i]$, etc.

In words: two elements are equivalent if they are in the same column - the class of an element is the column which contains it. Every element in $A$ is in only one column.

Answer 2

Consider an elementary school. Define an equivalence relation on the children in the school based on $\text{thing}_1 \sim \text{thing}_2$ if and only if they are in the same room. Then every child is assigned to a room and so every child lands in one and only one of the equivalence classes. Thus the set of children is the union of all of the equivalence classes (i.e. classrooms of children).

(I'm assuming none of the children are on a restroom break, in the nurse's office or in the principal's office for misbehavior.)

Answer 3

$xRy$ if and only if $x=y$ you can see $R$ is equivalence relation

Answer 4

2 examples I like:

1. Fix any natural number $n\geqslant 2$ and define $\sim$ on $\mathbb Z$ by $x\sim y$ if and only if $x$ and $y$ differ by a multiple of $n$ (this means that $x-y$ is a multiple of $n$). Then the equivalence class of $x$ is $[x]=\{\ x+kn\ |\ k\in\mathbb Z\ \}$, so $\mathbb Z$ is partitioned into the subsets of integers that have the same remainder in the Euclidean division by $n$. For $n=2$ for example, $x$ and $y$ are equivalent if they have the same parity and the equivalence classes are the sets of even numbers and the set of odd numbers.
2. Choose a surjective function $f$ from $\mathbb R$ onto $\mathbb R$. Then two real numbers $x$ and $y$ are equivalent if and only if $f(x)=f(y)$. What is $[x]$, the equivalence class of $x\in\mathbb R$ ? Let $a=f(x)$. Then $[x]=f^{-1}(\{a\})$. Therefore, the partition by equivalence class is the partition by the subsets that have the same value by $f$.

I hope it helps.