Closure Property

Closure Property is the property that deals with the operations and their results within a certain set of numbers. This property states that if a set of numbers is closed under any arithmetic operation like addition, subtraction, multiplication, and division, i.e. the operation is performed on any two numbers of the set the answer of the operation is in the set itself.

In this article, we will learn about Closure Property, Closure Property of Addition, Closure Property of Multiplication, Examples, and others in detail.

What is Closure Property?

When a set of numbers is given, an arithmetic operation is performed on those numbers then the outcome always belongs to the same set.

The image added below takes two integers a and b, and checks the closure property on all four basic arithmetic operations.

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Closure Property

Here, a and b are integers, and their sum, difference and multiplication all follow closure property for integers. But division of a and b is not necessarily a integer so division does not follow, closure property.

Closure Property Definition

When an arithmetic operation is performed to two numbers within a certain set of numbers, the final result always comes within the same set. Suppose a number set {5.10,15} is given. When you perform addition between 2 numbers of this set, the final number belongs to this set only.

Types of Closure Property

As you have learned the definition of closure property, it’s clear that arithmetic operations are performed within a number set. Now, we learn about the types of closure property.

Closure property is mainly divided into 4 parts:

• Closure Property of Addition
• Closure Property of Subtraction
• Closure Property of Multiplication
• Closure Property of Division

These four are the arithmetic operations we perform within a number set in closure property.

Closure Property of Addition

As you can understand with the name itself, you perform addition within the set and the resultant will come under the set only. For example, let’s take an even number set {2,4,6}. Take two numbers 2 & 4 from this set and perform addition on them. Here, 2+4= 6, the resultant is the same number as in the set.

Take another 2 numbers 4+6= 10, the resultant is not an even number. Here, this is violating the closure property.

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 Subtraction

Under this closure property, you perform subtraction within the set of numbers and the resultant will come under the set in the same way. For example, a number set {5,10,15} is given. Take 2 numbers 15 & 5 from this set and perform subtraction on them. Here, 15-5= 10, the outcome is under the set. Therefore it is following the closure property.

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 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

In closure under multiplication, you will multiply within the numbers in the set. Suppose take an even number set {2,4,8}. Multiply two numbers 2×4 = 8. It is an even number that’s why following the closure property. While we perform multiplication 8×2 = 16, it’s not an even number and not falling under the number set.

Closure Property of Multiplication
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 Division

For this type, let’s take a set {8,16,24}. Take 2 numbers 16 & 8. Now we divide 16÷8= 2. Here, 2 is within the original number set. The closure property is satisfied here because 2 is falling under the same set where we perform division.

Closure Property Formula

If we take two numbers a, and b from a set S then closure property formula states that, a (operator) b also belongs to set S. This is explained as,

“∀ a, b ∈ S ⇒ a (operator) b ∈ S“

Real numbers are closed under Addition, Multiplication and Subtraction operation but not division operation because, a/b is not a real number when b is zero.

Closure Property Examples

Now, you have understood what is closure property and what kind of arithmetic operations it includes. Now, it’s time to deeply understand the concept of closure property via some real life examples. Below are some examples of closure property with various kind of number:

Closure Property of Real Numbers

Real numbers also follows closure property. But first you need to understand what are real numbers.

Real numbers are the set of both rational numbers and irrational numbers. The real numbers are closed under all four arithmetic operations i.e. addition, subtraction, multiplication and division. When you perform any arithmetic operations on real numbers, the final result will also be a real number.

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Let’s understand the closure property of real numbers with a simple example:

Suppose a number set {1, 2, 3} is given. Let’s perform each arithmetic operation on this set to know whether it falls under closure property or not.

• Add 1 + 2 = 3, here 3 is also a real number.
• Subtract 3 – 2 = 1, here 1 is also a real number.
• Multiply 2 × 3 = 6, here 6 is also a real number.

• Divide 3 ÷ 0 = undefined {not a real number, and hence real number does not follow closure property under divide operator}

All the final results are real number and under the set. Therefore, closure property of real numbers is satisfied under Addition, Subtraction and Multiplication operation.

Closure Property of Rational Numbers

Rational numbers also a follower of closure property. Now, the real question is what are rational numbers.

Rational numbers includes fraction and decimal numbers. When two rational numbers are added, subtracted or multiplied the resultant will always be a rational number. Here, the division is an exception because the division by zero is infinite. So the final answer will not always be a rational number while performing division.

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Let’s take an example:

A number set {½, ⅔, ¾} is given.

• Add ½ + ⅔ = 7/6, which is also a rational number.
• Subtract ¾ – ½ = ¼. which is also a rational number.
• Multiply ½ × ⅔ = ⅓, which is also a rational number.
• Divide ¾ ÷ 0 = undefined {not a real number, and hence real number does not follow closure property under divide operator}.

Closure Property of Integers

Integers also follow closure property under various arithmetic operations. Integers include both positive and negative numbers of a number scale. When you perform any arithmetic operations on integers, the final answer will always be an integer.

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Let’s take a simple example:

A number set of integers {-3,0,3} is given.

• Add -3 + 3 = 0, which is an integer.
• Subtract 3 – 0= 3, which is an integer.
• Multiply 3 × 0 = 0, which is an integer.
• Divide -3 ÷ 0 = undefined {not a real number, and hence real number does not follow closure property under divide operator}.

Closure Property in Modular Arithmetic

When you add, subtract, multiply or take reminders within a modulus, the result stays congruent to the set modulus. This property is called closure property in modular arithmetic.

Closure Property in Group Theory

Group theory is the theory in which you deal with abstract algebraic structures.

Closure property is one of the fundamental properties which defines a group. When you perform any operation on elements within that group, the resultant always comes in the same group.

Closure Property of Addition

In group theory, the Closure Property is the most basic property of the set. When you do addition within a set, if the sum of any two elements also belongs to that set, it shows closure property under addition. For example,

Consider a number set N (Natural Numbers) under addition

• 2 + 4 = 6 ϵ N
• 4 + 6 = 8 ϵ N

Thus, closure property is followed under Addition operation.

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Read More,

• Properties of Rational Numbers
• Properties of Real Numbers
• Properties of Integers

Closure Property Examples

Example 1: Consider the set {1, 2, 3, 4, 5}. Is this set closed under addition?

Solution:

Yes, it’s closed. For instance, 2 + 3 = 5, which is still within the set.

Example 2: Given a set {6, 12, 18}. Is it closed under multiplication?

Solution:

Yes, it’s closed. Because, 6 * 18 = 108, which is also in the set.

Example 3: Is the set {1, 3, 5} forms a group under addition (mod 6).

Solution:

1 + 3 = 4 (in the set)

3 + 5 = 2 (in the set)

1 + 5 = 0 (in the set)

Hence, the set {1, 3, 5} under addition (mod 6) forms a group due to closure.

Example 4: Compute if {0, 4, 8} is closed under addition (mod 10).

Solution:

0 + 4 = 4 (in the set)

4 + 8 = 2 (violates closure as 2 is not in the set)

Hence, {0, 4, 8} is not closed under addition (mod 10).

Example 5: Verify the closure property of {1, 4, 7} under addition (mod 5).

Solution:

1 + 4 = 0 (in the set)

4 + 7 = 1 (in the set)

1 + 7 = 3 (violates closure as 3 is not in the set)

Therefore, {1, 4, 7} is not closed under addition (mod 5).

Closure Property – Practice Questions

Q1: Find if the set {2, 4, 6, 8} is closed under addition (mod 7).

Q2: Show if the set {3, 9, 15, 21} forms a group under multiplication (mod 10).

Q3: Verify closure in the set {1, 5, 9, 13} under addition (mod 12).

Q4: Determine if the set {0, 5, 10, 15} is closed under multiplication (mod 20).

Q5: Find if closure in the set {4, 9, 14, 19} under addition (mod 25).

Closure Property – FAQs

1. Why is Closure Property Important?

It helps us predict resultants of operations within certain sets and forms the basic of various mathematical concepts.

2. Are All Sets Closed Under Every Operation?

No, some sets might not be closed under specific arithmetic operations.

3. How Can I Verify Closure Property in a Set?

To check closure, perform the arithmetic operation (addition, multiplication, etc.) on each pair of elements in the set and verify if the result stays within the set.

4. What is Closure Property in Maths?

Closure property in Mathematics is the basic property that tells us that if two numbers from a set operates under a certain operator and the result is also in the same set then i follows the closure property under the operator.

5. What is the Formula for Closure Property?

Closure property formula states that, for two numbers a, and b from set N (natural numbers) then,

• a + b ∈ ℕ
• a × b ∈ ℕ
• a – b ∉ ℕ
• a ÷ b ∉ ℕ