Closure Property

Closure property states that when a set of numbers is closed under any arithmetic operation such as addition, subtraction, multiplication, and division, it means that when the operation is performed on any two numbers of the set with the answer being another number from the set itself. For example, the set of integers is closed with respect to addition/subtraction/multiplication but it is NOT closed with respect to division.

The closure property is important in many areas of mathematics, including algebra, group theory, ring theory, etc. Let us learn more about this property.

The closure property is defined as follows: When a given operation is performed on any two numbers from a given set and the result obtained is also present in the same set itself, the given set is said to be closed with respect to that particular operation. For example, the sum of any two natural numbers is again a natural number and hence the set of natural numbers is closed with respect to addition. However, the set of natural numbers is NOT closed with respect to subtraction as the difference of two natural numbers (example: 3 - 5 = -2) need not be a natural number.

The closure property is applicable for addition and multiplication for most of the number systems. In spite of that, for subtraction and division, some sets are not closed. Let us see the closure property of each arithmetic operation in brief.

The closure property of addition states that when any two elements of a set are added, their sum will also be present in that set. The closure property formula for addition for a given set S is: ∀ a, b ∈ S ⇒ a + b ∈ S.

Here are some examples of sets that are closed under addition:

• Natural Numbers (ℕ): ∀ a, b ∈ ℕ ⇒ a + b ∈ ℕ
• Whole Numbers (W): ∀ a, b ∈ W ⇒ a + b ∈ W
• Integers (ℤ): ∀ a, b ∈ ℤ ⇒ a + b ∈ ℤ
• Rational Numbers (ℚ): ∀ a, b ∈ ℚ ⇒ a + b ∈ ℚ

The set of non-zero integers (ℤ - {0}) is NOT closed under addition because -1 and 1 are in ℤ - {0} but their sum -1 + 1 = 0 ∉ ℤ - {0}.

The closure property of multiplication states that when any two elements of a set are multiplied, their product will also be present in that set. The closure property formula for multiplication for a given set S is: ∀ a, b ∈ S ⇒ a × b ∈ S.

Here are some examples of sets that are closed under multiplication:

• Natural Numbers (ℕ): ∀ a, b ∈ ℕ ⇒ a × b ∈ ℕ
• Whole Numbers (W): ∀ a, b ∈ W ⇒ a × b ∈ W
• Integers (ℤ): ∀ a, b ∈ ℤ ⇒ a × b ∈ ℤ
• Rational Numbers (ℚ): ∀ a, b ∈ ℚ ⇒ a × b ∈ ℚ

The set of irrational numbers is NOT closed under multiplication as the product of two irrational numbers doesn't need to be irrational. For example, √8 × √2 = √16 = 4, which is NOT irrational.

The closure property of subtraction states that when any two elements of a set are considered, their difference will also be present in that set. The closure property formula for subtraction for a given set S is: ∀ a, b ∈ S ⇒ a - b ∈ S.

Here are some examples of sets that are closed under subtraction:

• Integers (ℤ): ∀ a, b ∈ ℤ ⇒ a - b ∈ ℤ
• Rational Numbers (ℚ): ∀ a, b ∈ ℚ ⇒ a - b ∈ ℚ

Here are some examples of sets that are NOT closed under subtraction along with a counter-example.

• Natural numbers set is NOT closed under subtraction. Example: 1, 2 ∈ ℕ but 1 - 2 = -1 ∉ ℕ
• Whole numbers set is NOT closed under subtraction. Example: 0, 5 ∈ W but 0 - 5 = -5 ∉ W

The closure property of division states that when any two elements of a set are divided, the quotient will also be present in that set. The closure property formula for division for a given set S is: ∀ a, b ∈ S ⇒ a ÷ b ∈ S. Usually, most of the sets (including integers and rational numbers) are NOT closed under division. Here are some examples.

Here are some examples of sets that are NOT closed under division along with a counter-example.

• Integers set is NOT closed under division. Example: 2, 3 ∈ ℤ but 2 ÷ 3 = 2/3 ∉ ℤ
• Rational numbers set is NOT closed under division. Example: 1, 0 ∈ ℚ but 1÷ 0 ∉ ℚ

The closure property formula says "∀ a, b ∈ S ⇒ a (operator) b ∈ S", where

• S is a given set
• "operator" stands for any arithmetic operator indicating addition, subtraction, multiplication, division, etc.

We know that the set of real numbers is closed under each arithmetic operation. So if two real numbers a and b are given, then the closure property formula for the set of real numbers is given as follows:

Important Notes on Closure Property:

• Usually, the closure property of addition and multiplication are applicable to most sets like natural numbers, integers, whole numbers etc.
• But the closure property of subtraction and division are NOT applicable to most of the sets like natural numbers, whole numbers, etc.

☛Related Topics:

Listed below are a few topics that are related to closure property.

• Properties of Whole Numbers
• Properties of Natural Numbers
• Properties of Integers
• Properties of Rational Numbers

What is Closure Property in Maths?

The closure property states that for a given set and a given operation, the result of the operation on any two numbers of the set will also be an element of the set. Here are some examples of closed property:

• The set of whole numbers is closed under addition and multiplication (but not under subtraction and division)
• The set of rational numbers is closed under addition, subtraction, and multiplication (but not under division)

What is the Closure Property of Integers Under Addition and Subtraction?

For the set of integers:

• The closure property under addition says the sum of any two integers is also an integer.
• The closure property under subtraction says the difference between any two integers is also an integer.

Is the Closure Property of Subtraction Applicable Always?

No, it is not applicable always. The closure property of subtraction is NOT applicable for natural numbers and whole numbers. So we say that these sets are NOT closed under subtraction.

For Which Operations Closure Property is Applicable for Whole Numbers?

The closure property for whole numbers is applicable only with respect to the operations of addition and multiplication. For example, consider whole numbers 7 and 8, 7 + 8 = 15 and 7 × 8 = 56. Here 15 and 56 are whole numbers as well. This property is not applicable to subtraction and division.

Are Natural Numbers Closed Under Subtraction?

The closure property of natural numbers is applicable only for addition and multiplication. The subtraction of two natural numbers does not necessarily create another natural number, for example, 5 - 11 = -6. Thus, natural numbers set is not closed under subtraction.

What is the Closure Property of Rational Numbers with Respect to Addition?

The closure property of rational numbers with respect to addition states that when any two rational numbers are added, the result of all will also be a rational number. For example, consider two rational numbers 1/3 and 1/4, their sum is 1/3 + 1/4 = (4 + 3)/12 = 7/12, 7/12 is a rational number.

What is the Formula for Closure Property?

The closure property formula says that a given set with a given operation is closed only when the results of the operation on every pair of numbers from the given set are also present in the set. For example, the following is the closure property formula for the set of natural numbers for any two natural numbers a and b:

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