Operation Closure Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

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

Solution

  1. 1
    Check if subtracting two whole numbers always gives a whole number.
  2. 2
    Try \(5 - 8 = -3\). Is \(-3\) a whole number? No — whole numbers are \(\{0, 1, 2, 3, \ldots\}\).
  3. 3
    Counterexample found: \(5 - 8 = -3 \notin \mathbb{W}\).
  4. 4
    Whole numbers are NOT closed under subtraction.

Answer

No — counterexample: \(5 - 8 = -3\)
One counterexample is enough to disprove closure. Since \(5 - 8 = -3\) and \(-3\) is not a whole number, the set is not closed under subtraction.

About Operation Closure

A set is closed under an operation if performing that operation on members of the set always produces a result that is also in the set. For example, integers are closed under addition.

Learn more about Operation Closure →

More Operation Closure Examples

Example 1 easy

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

Example 3 easy

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

Example 4 medium

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

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