Closed Under

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Formal Definition

Let $*$ be a binary operation on $S$ and let $H$ be a subset of $S$. The subset $H$ is closed under $*$ if for all $a,b\in\ H$ we also have $a*b \in\ H$. In this case, the binary operation on $H$ given by restricting $*$ to H is the induced operation of $*$ on $H$.

Informal Definition

A set is closed under an operation if performance of that operation on two elements of the set produces a member of the same set.


The set $\mathbb {R}$ is closed under addition because for any $a,b\in \mathbb{R}$, $a+b=c$, where $c$ is also an element in $\mathbb{R}$.


Our usual addition + on the set $\mathbb{R}$ of real numbers does not induce a binary operation on the set $\mathbb{R^*}$ of nonezero real numbers because $2 \in\ \mathbb{R^*}$ and $-2 \in\ \mathbb{R^*}$, but $2 + (-2) = 0$ and $0 \notin\ \mathbb{R^*}$. Thus $\mathbb{R^*}$ is not closed under $*$.

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