Paper 4, Section I, $2 \mathrm{D}$

What is an equivalence relation on a set $X$ ? If $R$ is an equivalence relation on $X$, what is an equivalence class of $R$ ? Prove that the equivalence classes of $R$ form a partition of $X$.

Let $R$ and $S$ be equivalence relations on a set $X$. Which of the following are always equivalence relations? Give proofs or counterexamples as appropriate.

(i) The relation $V$ on $X$ given by $x V y$ if both $x R y$ and $x S y$.

(ii) The relation $W$ on $X$ given by $x W y$ if $x R y$ or $x S y$.

*Typos? Please submit corrections to this page on GitHub.*