# Set That Is Closed Under an Operation

## Set That Is Closed Under an Operation

Set in which an internal composition law.

### Examples

• The set of whole numbers is closed under addition and multiplication.
Any sum or product of whole numbers is a whole number.
Let n ∈ $$\mathbb{N}$$ and p ∈ $$\mathbb{N}$$.  Then : ∀n, p ∈ $$mathbb{N}$$ : (n + p) ∈ $$\mathbb{N}$$ and (n × p) ∈ $$\mathbb{N}$$.
• The set of whole numbers is not closed under subtraction and division.
Not all differences and quotients of whole numbers are whole numbers.
Let n ∈ $$\mathbb{N}$$ and p ∈ $$\mathbb{N}$$.  Then : ∃n, p ∈ $$\mathbb{N}$$ : (np) ∉ $$\mathbb{N}$$ and (n ÷ p) ∉ $$\mathbb{N}$$.