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.
Best Answer
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 $\{1,2,3,4\}$