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 ∈ [latex]\mathbb{N}[/latex] and p ∈ [latex]\mathbb{N}[/latex].  Then : ∀n, p ∈ [latex]mathbb{N}[/latex] : (n + p) ∈ [latex]\mathbb{N}[/latex] and (n × p) ∈ [latex]\mathbb{N}[/latex].
• 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 ∈ [latex]\mathbb{N}[/latex] and p ∈ [latex]\mathbb{N}[/latex].  Then : ∃n, p ∈ [latex]\mathbb{N}[/latex] : (n – p) ∉ [latex]\mathbb{N}[/latex] and (n ÷ p) ∉ [latex]\mathbb{N}[/latex].