Operation Closure Math Example 4

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

Example 4

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

Solution

  1. 1
    Test: \(7 \div 2 = 3.5\). Is 3.5 an integer? No.
  2. 2
    Counterexample: \(7 \div 2 = 3.5 \notin \mathbb{Z}\).
  3. 3
    Integers are NOT closed under division.

Answer

No — counterexample: \(7 \div 2 = 3.5\)
Dividing two integers can give a non-integer (like 3.5), so integers are not closed under division.

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 2 medium

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

Example 3 easy

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

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