Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 4 / 2C / Solution

An equivalence relation on a set $X$ is a binary relation that is reflexive, symmetric and transitive: every $x\in X$ satisfies $x\sim x$; $x\sim y$ implies $y\sim x$; and $x\sim y$, $y\sim z$ imply $x\sim z$. Each of these quantifiers ranges over the entire specified set. The separate entries below test these three properties.