Let S ={a, b, c, d, e}, and P={{a, b},{c, d},{e}}. (a) Verify that P really is a partiton of S. (b) Find the equivalence relation R on S induced by P.

Let $S =\{a, b, c, d, e\}$ , and $P=\{\{a, b\},\{c, d\},\{e\}\}.$

(a) By defenition, a partition of a set $S$ is a set of non-empty subsets of $S$  such that every element $x$  in $S$ is in exactly one of these subsets. Since $\{a, b\},\{c, d\},\{e\}$ are non-empty set and each element $s\in S$ is in exactly one of the sets $\{a, b\},\{c, d\}$ and $\{e\}$, $P$ really is a partiton of $S$.

(b) Let us find the equivalence relation $R$ on $S$ induced by $P$. By defenition, $(x,y)\in R$ if and only if $x$ and $y$ are elements of the same set of a partition. In our case, $R=\{(a,a),(a,b),(b,a),(b,b), (c,c),(c,d),(d,c),(d,d),(e,e)\}.$