As a refresher, if $R$ is an equivalence relation over a set $A$ and $x \in A$, then the equivalence class of $\boldsymbol{x}$ in $\boldsymbol{R}$, denoted $[x]_R,$ is the set
$$
[x]_R=\{y \in A \mid x R y\}
$$
Let's now introduce some new notation. If $R$ is an equivalence relation over a set $A,$ the index of $R,$ denoted $\boldsymbol{I}(\boldsymbol{R}),$ is the number of equivalence classes in $R$. Additionally, the width of $R,$ denoted $\boldsymbol{W}(\boldsymbol{R}),$ is the number of elements in the largest equivalence class in $R$.
If $R$ is an equivalence relation over a set $A$ and $|A|=n^2+1$ for some positive natural number $n$, then which of the following must be true?
- $\mathrm{I}(\mathrm{R}) \geq n+1$ and $\mathrm{W}(\mathrm{R}) \geq n+1$.
- $\mathrm{I}(\mathrm{R}) \leq n$ and $\mathrm{W}(\mathrm{R}) \leq n$
- $\mathrm{I}(\mathrm{R}) \geq n+1$ or $\mathrm{W}(\mathrm{R}) \geq n+1$ (or both).
- $\mathrm{I}(\mathrm{R})=n / 2$ and $\mathrm{W}(\mathrm{R})=n / 2$
