Closure property means: when you perform an operation (like addition or multiplication) on any two elements of a set, the result is still in that same set.

Simple definition

  • A set is closed under an operation if:
    • You take any two elements from the set.
    • Apply the operation (say +++, −-−, ×\times ×, or ÷\div ÷).
    • The answer you get is also in that set.
  • If you can find even one example where the result goes outside the set, then the set is not closed under that operation.

Easy examples

  • Natural numbers under addition:
    • 3+5=83+5=83+5=8. All are natural numbers, so naturals are closed under addition.
  • Natural numbers under subtraction:
    • 3−5=−23-5=-23−5=−2. −2-2−2 is not a natural number, so naturals are not closed under subtraction.
  • Even numbers under addition:
    • 10+12=2210+12=2210+12=22, still even, so even numbers are closed under addition.
  • Even numbers under division:
    • 100÷4=25100\div 4=25100÷4=25, which is not even, so even numbers are not closed under division.

General “formula” way

  • For a set SSS and an operation ∘\circ ∘:
    • If for all a,b∈Sa,b\in Sa,b∈S, we have a∘b∈Sa\circ b\in Sa∘b∈S, then SSS is closed under ∘\circ ∘.

In short, closure property asks: “Whenever I combine elements of this set with this operation, do I always stay inside the set?” If yes, the set is closed under that operation.