Closure Property: Definition, Formula, Examples

What Is Closure Property?

The closure property states that if a set of numbers (integers, real numbers, etc.) is closed under some operation (such as addition, subtraction, or multiplication, etc.), then performing that operation on any two numbers in the set results in the element belonging to the set.

To understand this, let’s look at some questions that we often wonder about.

• Is the sum of two natural numbers a natural number? Yes!
• Is the subtraction of two integers always an integer? Yes!
• Do we always get an integer by the division of two integers? No!

The closure property is directly linked with such properties of any given set with respect to an operation. Let’s explore the closure property in detail for different operations and different sets.

Definition of Closure Property

The closure property states that when a set of numbers is closed under an arithmetic operation, performing the operation on any two numbers in the set always results in a number belonging to the same set of numbers.

If the operation results in even a single element outside the set, we can say that the set is not closed under the given operation.

Example 1: The addition of two real numbers is always a real number. Thus, real numbers are closed under addition.

Example 2: Subtraction of two natural numbers may or may not be a natural number. Thus, natural numbers are not closed under subtraction.

Closure Property of Real Numbers

The set of rational numbers and irrational numbers together form the set of real numbers.

Real numbers include:

• Natural numbers $\left\{1,\;2,\;3,\; …\right\}$
• Whole numbers $\left\{0,\;1,\;2,\;3,\; …\right\}$
• Integers $\left\{…,\;-\;3,\;-\;2,\;-\;1,\;0,\;1,\;2,\;3,\;….\right\}$
• Rational numbers 
• Irrational numbers 

Real numbers are closed under the following operations: addition, subtraction, multiplication, division. Note that division by 0 is not defined. Division by zero is the only case where the closure property does not apply for real numbers. We can say that the real numbers are closed under non-zero division

Closure Property of Real Numbers
Real number $+$ Real number $=$ Real number	Closed under addition
Real number $\;-\;$ Real number $=$ Real number	Closed under subtraction
Real number Real number $=$ Real number	Closed under multiplication
Real number Real number $=$ Not always a real number	Closed only under non-zero division

Closure Property for Integers

The set of integers is given by $Z = \left\{…,\; -\;3,\; -\;2,\; -\;1,\; 0,\; 1,\; 2,\; 3,\; … \right\}$.

The closure property holds true for addition, subtraction, and multiplication of integers. It does not apply for the division of two integers.

• Closure property of integers under addition:

The closure property of addition of integers states that the sum of any two integers will always be an integer. If a and b are any two integers, $a + b$ will be an integer.

Examples:  $(\;-\;5) + 8 = 3$

                    $10 + 6 = 16$

• Closure property of integers under subtraction:

The difference between any two integers will always be an integer. If a and b are any two integers, $a \;-\; b$ will also be an integer.

Examples: $10 \;-\; 6 = 4$

               $(\;-\;6) \;-\; (\;-\;3) = \;-\;3$

• Closure property of integers under multiplication:

The product of any two integers will be an integer. If a and b are any two integers, a b will also be an integer.

Examples: $3 \times (\;-\;9) = \;-\;27$

                $(\;–\;7) \times (\;-\;9) = 63$

• Closure property of integers under division:

If you divide an integer by another integer, you may or may not get an integer as a quotient. Also, the division by 0 is not defined.

Examples: $\;-\;10 \div 5 = \;-\; 2$ is an integer.

                 $(\;-\;5) \div ( \;-\;25) = \frac{1}{5}$ is not an integer

Closure Property of Integers
Integer $+$ Integer $=$ Integer	Closed under addition
Integer $-$ Integer $=$ Integer	Closed under subtraction
Integer Integer $=$ Integer	Closed under multiplication
Integer Integer $=$ Not always an integer	Not closed under division

Closure Property of Rational Numbers

Rational numbers are the numbers that can be represented in the form of $\frac{p}{q}$, where $q \neq 0$; p and q are integers.

The rational numbers are closed under addition, subtraction, and multiplication. The closure property isn’t applicable for division since division by zero isn’t defined.

In other words, we can say that closure property is applicable for division of rational numbers except division by zero.

1)  $\frac{4}{7} + \frac{2}{3} = \frac{26}{21}$  is a rational number

2)  $\frac{4}{3} \;–\; \frac{2}{4} = \frac{6}{12}$  is a rational number

3)  $\frac{3}{5} \times \frac{2}{3} = \frac{6}{15}$  is a rational number

Closure Property of Rational Numbers
Rational number $+$ Rational number $=$ Rational number	Closed under addition
Rational number $–$ Rational number $=$ Rational number	Closed under subtraction
Rational number Rational number $=$ Rational number	Closed under multiplication
Rational number $\div$ Rational number $=$ Not always a Rational number	Closed only under non-zero division

Closure Property of Whole Numbers

The set of whole numbers is given by $W = \left\{0,\; 1,\; 2,\; 3,\; 4,\; …\right\}$. Thus, whole numbers include natural numbers and 0.

The whole numbers are closed under addition and the multiplication. If a and b are two whole numbers, $a + b$ is a whole number and $a \times b$ is also a whole number.

For example,

1) $2 + 6 = 8$ …a whole number

2) $2 \times 6 = 12$ …a whole number

Whole numbers are not closed under subtraction and division. If a and b are two whole numbers, then $a \;-\; b$ and $a \div b$ is not always a whole number.

For example:  

1) $6 \;-\; 1 = 5$ is a whole number.

2) $1\;-\; 6 = \;-\; 5$  is not a whole number.

3) $\frac{12}{6} = 2$ is a whole number.

4)  $\frac{1}{6} = 0.167$ is a decimal number, not a whole number.

Closure Property of Whole Numbers
Whole number $+$ Whole number $=$ Whole number	Closed under addition
Whole number Whole number $=$ Whole number	Closed under multiplication
Whole number $-$ Whole number $=$ Not always a whole number	Not closed under subtraction
Whole number $\div$ Whole number $=$ Not always a whole number	Not closed under division

Closure Property of Addition

If the addition of two numbers in a given set of numbers belongs to the set, then we say that the given set of numbers is closed under addition. 

The set of real numbers, natural numbers, whole numbers, rational numbers, and integers are closed under addition.

Closure Property of Addition
Real Numbers	$a + b =$ Real number (a, b are real numbers.)
Rational Numbers	$a + b =$ Rational number (a, b are real numbers.)
Integers Numbers	$a + b =$ Integer (a, b are integers.)
Natural Numbers	$a + b =$ Natural number (a, b are natural numbers)
Whole Numbers	$a + b =$ Whole number (a, b are whole numbers)

Closure Property of Multiplication

If the multiplication or product of two numbers in a given set of numbers belongs to the set, then we say that the given set of numbers is closed under multiplication. 

The closure property of multiplication is applicable for natural numbers, whole numbers, integers, and rational numbers.

Closure Property of Multiplication
Real Numbers	$a \times b =$ Real number (a, b are real numbers.)
Rational Numbers	$a \times b =$ Rational number (a, b are rational numbers.)
Integers Numbers	$a \times b =$ Integer (a, b are integers.)
Natural Numbers	$a \times b =$ Natural number (a, b are natural numbers)
Whole Numbers	$a \times b =$ Whole number (a, b are whole numbers)

Closure Property of Subtraction

If the subtraction of two numbers in a given set of numbers belongs to the set, then we say that the given set of numbers is closed under subtraction. 

This property is applicable for real numbers, integers, and rational numbers.

Closure Property of Subtraction
Real Numbers	$a \;-\; b =$ Real number (a, b are real numbers.)
Rational Numbers	$a \;-\; b =$ Rational number (a, b are real numbers.)
Integers	$a \;-\; b =$ Rational number (a, b are real numbers

Formula for Closure Property

For two real numbers a and b, the closure property formula can be given as

$a + b = R$

$a \;-\; b = R$

$a \times b = R$

$a \div b \neq R$ …since the division by 0 is not defined

Similarly, you can write the formulas for different sets with respect to each operation.

Facts about Closure Property

Conclusion

In this article, we learned that the closure property is one of the most significant properties in mathematics. The closure property denotes that the set is complete. This indicates that any operation performed on elements within a set produces a result that belongs to the same set. Now let’s apply this knowledge to solve some closure property examples.

Solved Examples on Closure Property

1. Write whether the following statements are true or false.

a. Rational numbers are closed under division. 

b. Natural numbers are closed under division.

c. Whole numbers are not closed under subtraction and division.

Solution: 

a. Rational numbers are closed under division.

False 

Rational numbers are closed under addition, subtraction, and multiplication but not under division. 

b. Natural numbers are closed under division.

False

Consider two natural numbers 1 and 2.

$1\div2 = 0.5$, which is not a natural number.

c. Whole numbers are not closed under subtraction and division.

True

Whole numbers are only closed under addition and multiplication.

2. Are integers closed under division? Explain why or why not. 

Solution:

Integers are not closed under division.

Consider a counter example. 

The numbers 6 and 7 are integers.

However, $6\div 7 = 0.85$ is not an integer. 

3. Give a counterexample to support the statement: Whole numbers are not closed under subtraction. 

Solution: 

Whole numbers include natural numbers and 0. 

$W = \left\{0,\; 1,\; 2,\; 3,\; …\right\}$

For any two whole numbers a and b, the difference $a \;-\; b$ may or may not be a whole number.

Consider a counterexample.

0 and 1 are whole numbers. 

However, $0 \;-\; 1 = \;-\;1$ is not a whole number.

Practice Problems on Closure Property

Frequently Asked Questions on Closure Property