## Definitions

Let $S$ be a set.

• A relation on $S$ is a subset of $S×S$. Write ${s}_{1}\sim {s}_{2}$ if the pair $\left({s}_{1},{s}_{2}\right)$ is in the subset.
• A relation $\sim$ on $S$ is reflexive if $\sim$ satisfies $if s∈S then s∼s.$
• A relation $\sim$ on $S$ is symmetric if $\sim$ satisfies $if s1,s2∈S and s1∼s2 then s2∼s1 .$
• A relation $\sim$ on $S$ is transitive if $\sim$ satisfies $if s1,s2, s3∈S and s1∼s2 and s2∼s3 then s1∼s3 .$
• An equivalence relation on $S$ is a relation on $S$ that is reflexive, symmetric and transitive.

Examples. Let $S$ be the set $\left\{1,2,6\right\}$.

(a)   ${R}_{1}=\left\{\left(1,1\right),\left(2,6\right),\left(6,1\right)\right\}$ is a relation on $S$.
(b)   ${R}_{1}$ is not reflexive, not symmetric and not transitive.
(c)   ${R}_{2}=\left\{\left(1,1\right),\left(2,6\right),\left(6,1\right),\left(2,1\right)\right\}$ is a relation on $S$.
(d)   ${R}_{2}$ is transitive but not reflexive and not symmetric.

Let $S$ be a set.

• Let $\sim$ be an equivalence relation on $S$ and let $s\in S$. The equivalence class of $s$ is the set $[s] = {t∈S | t∼s}.$
• A partition of a set $S$ is a collection of subsets ${S}_{\alpha }$ such that
(a)   if $s\in S$ then there exists $\alpha$ such that $s\in {S}_{\alpha }$, and
(b)   if ${S}_{\alpha }\cap {S}_{\beta }\ne \varnothing$ then ${S}_{\alpha }={S}_{\beta }$.
(a)   Let $S$ be a set and let $\sim$ be an equivalence relation on $S$.
Then the set of equivalence classes of the relation $\sim$ is a partition of $S$.
(b)   Let $S$ be a set and let $\left\{{S}_{\alpha }\right\}$ be a partition of $S$.
Then the relation defined by $s∼t if s and t are in the same Sα$ is an equivalence relation on $S$.

Proposition (eqrelptn) shows that the concepts of an equivalence relation on $S$ and of a partition on $Sare essentially the same. Each equivalence relation onSdetermines a partition and vice versa.$

Example. Let $S=\left\{1,2,3,\dots ,10\right\}$. Let $\sim$ be the equivalence relation determined by

 $1\sim 5$, $2\sim 3$, $9\sim 10$, $1\sim 7$, $5\sim 8$, $10\sim 4$,
Since we are requiring that $\sim$ is an equivalence relation, we are assuming that we have all the other relations we need such that $\sim$ is reflexive, symmetric and transitive;
 $1\sim 1$, $2\sim 2$, … $9\sim 9$, $10\sim 10$, $5\sim 7$, $7\sim 8$, $7\sim 5$, $5\sim 1$, …
The equivalence classes are given by $ == = = {1,5,7,8}, = = {2,3},  = {6},and ==  = {4,9,10},$ and the sets $S1 ={1,5,7,8}, S2 ={2,3}, S3 = {6}, and S4 = {4,9,10}$ form a partition of $S$.

## Notes and References

These notes are an updated version of notes of Arun Ram from 1994.

## References

[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.