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 ∈ \(\mathbb{N}\) : (n – p) ∉ \(\mathbb{N}\) and (n ÷ p) ∉ \(\mathbb{N}\).