How can I find the partitions of an equivalence relation?

Question

I have the following equivalence relation:

$$\{(1,1),(1,4), (2,2), (3,3), (4,1), (4,4)\}$$

On the set: $ A = \{1,2,3,4\}$

How can I find it's partitions? This example will help me understand the more general process of finding partitions for equivalence relations. Thanks.

Answer 1

Denote by $$\rho=\{(1,1),(1,4),(2,2),(3,3),(4,1),(4,4)\}$$ your equivalence,if $(x,y)\in\rho$ then we write $x\rho y$ you have $1\rho1\rho4\rho4\rho1$ so first class of equivalence is $\{1,4\}$ then $2\rho2$ the second class is $\{2\}$ and $3\rho3$ the third class is $\{3\}$ definitely your equivalence defines a partition $$\{\{1,4\},\{2\},\{3\}\}$$ of set $\{1,2,3,4\}$

Answer 2

In addition to every element being equivalent to itself, $1$ and $4$ are equivalent to each other. $2$ is not equivalent to anything by itself and $3$ is not equivalent to anything but itself. Hence $1$ and $4$ are in a class together, and $2$ and $3$ are singletons.

But this is just one partition, not "the partitions [plural]".

If you have trouble with problems like this, try drawing a picture: \begin{align} & 1 \longleftrightarrow 4 \\ & 2 \\ & 1 \end{align}

Answer 3

We can also approach this as follows. Define a function $f:A\to S$ where $S$ is any set with at least four elements, which follows the rule $f(x)=f(y)$ if and only if $x$ and $y$ are equivalent. Then, the partition is just the set of nonempty preimages of $f$.

Here, say we have the set $S=\{w,x,y,z\}$, and define $f$ by $f(1)=w$ arbitrarily. Since $2$ and $1$ are not equivalent, define $f(2)=x$, some other element of $S$. $3$ is not equivalent to either $1$ or $2$, so $f(3)=y$ some unused element of $S$. Finally, $4$ is equivalent to $1$, so $f(4)=f(1)=z$.

Now, we list the preimages: $f^{-1}[w]=\{1,4\}$, $f^{-1}[x]=\{2\}$, $f^{-1}[y]=\{3\}$, and $f^{-1}[z]=\emptyset$. Thus our partition is given by the first three sets.