Closure Property

Definition

A set $S$ is said to have the closure property under a function $f$ if applying $f$ to elements of $S$ always produces an output that is also in $S$.

• If $f$ is an operation on $S$, then $S$ is automatically closed under $f$ (by definition).
• The concept of closure is more meaningful when $f$ is any function $f:X^n\to Y$ with $S\subseteq X$ and $Y\supseteq S$, and you want to check whether $f(S^n)\subseteq S$.

Examples

• $\mathbb{Z}$ is closed under addition and multiplication.
• $\mathbb{Z}$ is not closed under division: dividing two integers may produce a non-integer.

Key Links

• Related to Operation (Mathematics)
• Uses the ideas of Function, Domain, Codomain, and Image