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

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.

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: A set is closed under an operation when combining members always lands back inside that set.

Common stuck point: The procedure for operation closure is the easy part; the trap is declaring closure from a few examples. Asking "Does combining any two members of the set always give a result still in the set?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does combining any two members of the set always give a result 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.

Answer

Yes — whole numbers are closed under addition

First step

Closure means: performing the operation on two members of the set always gives another member of the same set.

Full solution

2
Test: $5 + 7 = 12$ . Is 12 a whole number? Yes.
3
Test: $12 + 0 = 12$ . Is 12 a whole number? Yes.
4
Whole numbers are closed under addition: the sum of any two whole numbers is always a whole number.

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.

Example 3

medium

Are the integers closed under exponentiation? Justify with a counterexample if not.

Example 4

medium

Is the set

\{-1, 1\}

closed under multiplication?

Example 5

hard

Decide whether the set of

2\times 2

integer matrices is closed under addition and under matrix multiplication.

Example 6

hard

Show that the set of odd integers is NOT closed under addition but IS closed under multiplication.

Example 7

challenge

Show that the set

\{z \in \mathbb{C} : |z| = 1\}

is closed under multiplication.

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.

Example 3

easy

Is the set of integers closed under addition? (yes/no)

Example 4

easy

Is the set of integers closed under division? (yes/no)

Example 5

easy

Are the natural numbers (

1,2,3,\dots

) closed under subtraction? (yes/no)

Example 6

easy

Are the even integers closed under addition? (yes/no)

Example 7

easy

Are the odd integers closed under addition? (yes/no)

Example 8

easy

Are the whole numbers closed under multiplication? (yes/no)

Example 9

easy

Is the set

\{0, 1\}

closed under multiplication? (yes/no)

Example 10

easy

Are the positive integers closed under addition? (yes/no)

Example 11

medium

Is the set of rational numbers closed under division (excluding division by zero)? Explain.

Example 12

medium

Give a counterexample showing the natural numbers are not closed under division.

Example 13

medium

Is the set of irrational numbers closed under addition? Give reasoning.

Example 14

medium

Is the set

\{-1, 0, 1\}

closed under multiplication? Check all products.

Example 15

medium

Are the multiples of 5 closed under addition? Justify.

Example 16

medium

Is division by zero a reason that nonzero rationals are 'almost' closed under division? Explain in one line.

Example 17

medium

Are the negative integers closed under multiplication? Justify.

Example 18

medium

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

Example 19

medium

Is the set

\{1\}

closed under multiplication? Under addition?

Example 20

challenge

Determine whether the set

\{0, 1, 2\}

under addition modulo 3 is closed.

Example 21

challenge

Is the set of perfect squares closed under multiplication? Justify.

Example 22

challenge

Show that the set of odd integers is closed under multiplication but not addition.

Example 23

easy

Is the set of integers closed under subtraction?

Example 24

easy

Are the multiples of 3 closed under addition?

Example 25

easy

Are the multiples of 3 closed under multiplication?

Example 26

easy

Is the set of even integers closed under multiplication?

Example 27

easy

Is the set of positive rationals closed under multiplication?

Example 28

easy

Are the negative integers closed under addition?

Example 29

medium

Are the irrational numbers closed under multiplication?

Example 30

medium

Is the set

\{0,1,2,3\}

closed under addition modulo 4?

Example 31

medium

Is the set of all real numbers closed under taking square roots?

Example 32

medium

Are the prime numbers closed under multiplication?

Example 33

medium

Are the powers of

2

(

\{1,2,4,8,\dots\}

) closed under multiplication?

Example 34

medium

Are the powers of

2

closed under addition?

Example 35

hard

Is the set

S = \{a + b\sqrt{2} : a, b \in \mathbb{Q}\}

closed under multiplication?

Example 36

hard

Are the integers

\geq 0

closed under the operation

a \circ b = a + b - ab

Example 37

hard

Is the set of polynomials with integer coefficients closed under multiplication?

Example 38

hard

Is the set of

n\times n

invertible real matrices closed under multiplication?

Example 39

challenge

Is the set

\{a + b\sqrt[3]{2} : a, b \in \mathbb{Q}\}

closed under multiplication?

Example 40

challenge

Is the set of all sequences of zeros and ones (bitstrings of fixed length

n

) closed under bitwise XOR?

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Related Concepts

Addition Subtraction Multiplication Division Algebra as Structure

Background Knowledge

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

additionsubtractionmultiplicationdivision