A [[Set|set]] $S$ is considered closed over an [[Operation|operation]] or [[Function|function]] $\mu$ if for any inputs $x_{0},x_{1},x_{2},\dots x_{i} \in S$, the [[Function|mapping]] $\mu$ leads to another element $y \in S$. $\huge a \circ b \in S $ $\huge \begin{align} \mu(x_{0},x_{1}, \cdots) \in S \end{align}$ >[!example] >Multiplication $\cdot$ is closed over the [[Integer|integers]] $\Z$, as multiplying any two integers will always produce an integer. However, division $\div$ is not closed as dividing two integers can produce a non-integer > > >$\huge \begin{align} >\forall a,b&\in \Z& \\ >a\cdot b &\in \Z \\ >\exists a,b\in \Z &: a\div b \notin \Z >\end{align}$ > >A simple counter example is the integers $1$ and $2$, as $1 \div 2$ is $\frac{1}{2}$ and is not an integer.