Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 4 / 2E / Solution

An equivalence relation is a binary relation that is simultaneously a reflexive relation, a symmetric relation and a transitive relation: $x\sim x$; $x\sim y$ implies $y\sim x$; and $x\sim y$, $y\sim z$ imply $x\sim z$. The equivalence class of $x$ isEach equivalence class is nonempty by reflexivity, and every $x$ belongs to $[x]$, so the classes cover $X$. If $z\in [x]\cap [y]$, then the symmetric relation and transitive relation properties give $x\sim z\sim y$. Every $w\in [x]$ therefore satisfies $w\sim x\sim y$, so $[x]\subseteq [y]$; the reverse inclusion follows similarly. Thus two classes are either identical or disjoint. The distinct equivalence classes consequently form a set partition of $X$.

The divisibility-comparability relation is a reflexive relation and a symmetric relation, but is not an equivalence relation. Indeed, $2\sim 6$ and $6\sim 3$, whereas neither $2\mid 3$ nor $3\mid 2$, so transitivity fails.

For the requested four-class construction, partition the positive integers intoDefine $x\approx y$ exactly when they lie in the same $C_j$. Equality of the class label is an equivalence relation. Its four classes are the two finite singletons and the two infinite parity classes displayed above.