Closure (mathematics)

Closure (mathematics)

In mathematics, a <a href="/en/Subset" title="Subset" class="wl">subset</a> of a given <a href="/en/Set_(mathematics)" title="Set (mathematics)" class="wl">set</a> is closed under an <a href="/en/Operation_(mathematics)" title="Operation (mathematics)" class="wl">operation</a> of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the <a href="/en/Natural_number" title="Natural number" class="wl">natural numbers</a> are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are. 

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Operation on the subsets of a set</div>
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Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.

The closure of a subset is the result of a <a href="/en/Closure_operator" title="Closure operator" class="wl">closure operator</a> applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example <a href="/en/Linear_span" title="Linear span" class="wl">linear span</a>) or the generated set.

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.  Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span or the generated set. 