Operation Closure Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Operation Closure.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

When an operation on elements of a set always produces an element in the same set.

Adding two whole numbers always gives a whole number—closed under addition.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Closure tells us whether we stay within a number system after an operation.

Common stuck point: Checking closure requires testing all possible inputs, not just examples.

Sense of Study hint: Try a counterexample: pick two numbers from the set, perform the operation, and check if the result is still in the set.

Worked Examples

Example 1

easy
Are whole numbers closed under addition? Test with \(5 + 7\) and \(12 + 0\). Explain what closure means.

Solution

  1. 1
    Closure means: performing the operation on two members of the set always gives another member of the same set.
  2. 2
    Test: \(5 + 7 = 12\). Is 12 a whole number? Yes.
  3. 3
    Test: \(12 + 0 = 12\). Is 12 a whole number? Yes.
  4. 4
    Whole numbers are closed under addition: the sum of any two whole numbers is always a whole number.

Answer

Yes — whole numbers are closed under addition
A set is closed under an operation if applying the operation to any elements of the set always produces another element in the set.

Example 2

medium
Are whole numbers closed under subtraction? Provide a counterexample if not.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Are integers closed under multiplication? Test with \((-3) \times 4\) and \((-2) \times (-5)\).

Example 2

medium
Are integers closed under division? Provide a counterexample if not.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

additionsubtractionmultiplicationdivision