Given a set $S=\left\{1,2,3\right\}$ and two equivalence relations $\sim_B$ and $\sim_A$ on $S$, such that,
$\sim_A$ is defined as : $$\sim_A=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right)\right\}$$
Equivalence classes of $S$ with respect to $\sim_A$ are : $$\left[1\right]_{\sim_A}=\left\{1,2\right\}$$$$\left[2\right]_{\sim_A}=\left\{1,2\right\}$$$$\left[3\right]_{\sim_A}=\left\{3\right\}$$
$\sim_B$ is defined as : $$\sim_B=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right),\left(1,3\right),\left(3,1\right),\left(2,3\right),\left(3,2\right)\right\}$$
Equivalence classes of $S$ with respect to $\sim_B$ are :
$$\left[1\right]_{\sim_B}=\left\{1,2,3\right\}$$$$\left[2\right]_{\sim_B}=\left\{1,2,3\right\}$$$$\left[3\right]_{\sim_B}=\left\{1,2,3\right\}$$
The corresponding partitions are:
$P_{\sim_A}=\left\{\left\{1,2\right\},\left\{3\right\}\right\}$
$P_{\sim_B}=\left\{\left\{1,2,3\right\}\right\}$
Here every element in $P_{\sim_A}$ is a subset of the single element in $P_{\sim_B}$ and the single element is a union of some classes in $P_{\sim_A}$
Also clearly the equivalency of every two elements contained in $S$ under ${\sim_A}$ implies their equivalency under${\sim_B}$ so I think $P_{\sim_A}$ is a refinement of $P_{\sim_B}$ or equivalently $\sim_A$ is finer than $\sim_B$.
Am I right?
Can someone gives me a better examples of finer/coarser relation?