What is Closure Property in Maths?

A set is closed for an operation in mathematics when we can apply and complete that operation on the elements of the set and always receive an element from the same set. As a result, a set either has or does not have closure for a particular mathematical operation. In general, we can say that a set that is closed under an operation or set of functions is said to satisfy the closure property. A closure property is usually introduced as a hypothesis, which is known as the axiom of closure.

Here, we will learn what is closure property, the closure property formula and related concepts with the help of a few solved problems.

Closure Property Statement

The closure property in maths states that if we add or multiply any two real numbers, we will get only one unique answer and that answer will also be a real number.

Let’s go through some closure property examples.

The following are some important instances of closure properties.

• In set theory, the transitive closure of a set is an example of the closure property.
• The transitive closure of a binary relation is an example of closure property in set theory.

The closure property formula of addition and multiplication is discussed below.

Closure Property of Addition

The best way to demonstrate the closure property of addition is to use real numbers. When we add two real numbers, we get another real number since the set of real numbers is closed under addition. There will be no way to get anything else (suppose a complex number) than a real number here.

6 + 13 = 19

$\frac{1}{3}$ + $\frac{5}{2}$ = $\frac{17}{6}$

6.24 + 7.5 = 13.74

$5{\sqrt{5}} - 2{\sqrt{5}} = 3{\sqrt{5}}$

Closure Property under Multiplication

Real numbers are closed when they are multiplied because the product of two real numbers is always a real number. Natural numbers, whole numbers, integers, and rational numbers all have the closure property of multiplication.

Examples:

$ 8 \times 0$ = 0

${\frac{3}{4}} \times {- \frac{1}{2}} = {- \frac{3}{8}}$

$ {\sqrt{3}} \times {\sqrt{5}} = {\sqrt{15}}$

$ {- 11} \times {-3}$

Closure Property for Integers

The closure property holds true for integer addition, subtraction, and multiplication.

Closure Property of Integers Under Addition

Any two integers added together will always be an integer, i.e., if a and b are two integers, (a + b) will be an integer.

Example:

(-8) + 6 = 2

11 + 9 = 20

Closure Property of Integers Under Subtraction

Any difference between two integers will always be an integer, i.e., if a and b are both integers, (a – b) will always be an integer.

Example:

19 – 6 = 13

-6 – (-3) = -3

Closure Pproperty of Integers Under Multiplication

The product of any two integers will also be an integer, i.e., if a and b are two integers, ab will be an integer as well.

Example:

3 × (-9) = -27

(–7) × (-9) = 63

Closure Property of Integers Under Division

Since the quotient of any two numbers a and b may or may not be an integer, the division of integers does not follow the closure property. The quotient can be indeterminate at times (when the divisor is 0).

Example:

-10 ÷ 2 = -5 (it is an integer)

(−4) ÷ (−16) = $\frac{1}{4}$ (it is not an integer)

Closure Property of Rational Numbers

The closure property is true for rational number addition, subtraction, and multiplication.

Closure Property of Rational Numbers Under Addition

Any two rational numbers added together will always be a rational number, i.e., if a and b are both rational numbers, a + b will always be a rational number.

Example:

$\frac{5}{6} + \frac{2}{3} = \frac{3}{2}$

$-\frac{1}{2} + \frac{1}{4} = -\frac{1}{4}$

Closure Property of Rational Numbers Under Subtraction

Any difference between two rational numbers will always be a rational number, i.e., if a and b are both rational numbers, a – b will always be a rational number.

Example:

$\frac{7}{8} - \frac{3}{8} = \frac{1}{2}$

$\frac{6}{7} - (-\frac{3}{7}) = \frac{9}{7}$

Closure Property of Rational Numbers Under Multiplication

The closure property of multiplication states that the product of any two rational numbers will be a rational number, i.e., if a and b are both rational numbers, ab will be as well.

Example:

$\frac{3}{2} \times \frac{2}{9} = \frac{1}{3}$

$-\frac{7}{4} \times \frac{5}{2} = -\frac{35}{8}$

Closure Property of Rational Numbers Under Division

The quotient of any two rational numbers, a and b, may or may not be a rational number, the division of rational numbers do not follow the closure property. That is, if we set the value of b to 0, the result of a/b will be undefined.

Closure Property of Whole Numbers

The closure property holds true for whole number addition and multiplication. Subtraction and division are not allowed.

Closure Property of Whole Numbers Under Addition

If we take the sum of any two whole numbers, it will always be a whole number, i.e., consider a and b are any two whole numbers, then their addition (a + b) will be a whole number.

Example:

12 + 0 = 12

9 + 7 = 16

Closure Property of Whole Numbers Under Multiplication

The product of any two whole numbers will be a whole number, i.e., consider a and b are any two whole numbers, then the product (ab) will also be a whole number.

Example:

4 × 6 = 24

0 × 7 = 0

Closure Property of Whole Numbers Under Subtraction

If we find the difference between any two whole numbers the result may or may not be a whole number. Hence, we can say the whole numbers are not closed under subtraction.

Example:

13 – 15 = -2 (it is not a whole number)

4 – 0 = 4 (it is whole number)

Closure Property of Whole Numbers Under Division

Division of whole numbers doesn’t follow the closure property because the quotient of any two whole numbers a and b, may or may not be a whole number.

Example:

18 ÷ 4 = $\frac{9}{2}$ (it is not a whole number)

$\frac{10}{2}$ = 5 (it is a whole number)

Solved Examples

1. Help Rosie to find if 17 ÷ 2 belongs to the closure property.

Solution: The given numbers 17 and 2 belongs to natural numbers. When the given operation is performed then the result obtained is,

17 ÷ 2 = 8.5

8.5 does not belong to natural numbers. Hence, closure property does not apply here.

2. Prove if this equation comes under the closure property $\frac{1}{3} - \frac{1}{4}$.

Solution: According to the closure property, the difference or the output obtained will be a rational number if any of the two rational numbers on a number line are subtracted from each other.

$\frac{1}{3} - \frac{1}{4} = \frac{(4-3)}{12} = \frac{1}{12}$

$\frac{1}{12}$ is a rational number. Hence, the given expression accepts closure property.

Practise Questions

1. Which among the following properties is not affected by the sum, if the order of the given integers is changed?

1. Commutative property

2. Closure property

3. Associative property

4. Distributive property

Ans: Option a

2. Polynomials are closed under subtraction and addition

1. True

2. False

Ans: Option a

Conclusion

From the above examples, we can conclude that the closure property means that a set is closed for some mathematical operation. We have discussed the closure property of integers, rational numbers, and whole numbers. We have seen that the closure property of the whole numbers is only applicable for addition and multiplication. It should be noted that the closure property of rational numbers holds true for addition, multiplication and subtraction.