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) ∉ \(\mathbb{N}\).

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