Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 2E (Numbers and Sets)

A relation on $S$ is a subset of $S\times S$. An equivalence relation is reflexive, symmetric, and transitive. Its equivalence class at $x$ is $[x]=\{y\in S:x\sim y\}$.

This is not an equivalence relation because it is not reflexive: $0\nsim 0$, since $0\cdot 0\ngtr 0$.

The relation is reflexive and symmetric, but not transitive. For example,while $1\nsim -1$. Hence it is not an equivalence relation.

This is an equivalence relation: $x-x=0\in \mathbb{Z}$; if $x-y\in \mathbb{Z}$, then $y-x\in \mathbb{Z}$; and integer differences add. Its classes arethe cosets of $\mathbb{Z}$ in $\mathbb{R}$.

This relation is reflexive and transitive, but not symmetric: $1\sim 0$ whereas $0\nsim 1$. It is therefore not an equivalence relation.