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.

What is Closure 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.

Closure Property of Addition

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 - ) is NOT closed under addition because -1 and 1 are in ℤ - but their sum -1 + 1 = 0 ∉ ℤ - .

Closure Property of Multiplication

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.

Closure Property of Subtraction

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

Closure Property of Division

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

Closure Property Formula

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

Examples of Closure Property

Example 1: Does the set of rational numbers satisfy the closure property of division? If not, provide a counter-example. Solution: The set of rational numbers is NOT closed under division. This is because, for the rational numbers -3 and 0, their quotient is -3/0 (which is not at all defined) is NOT present in the rational numbers set. Answer: Not closed.

Example 2: Is the set of integers closed under subtraction? Justify your answer. Solution: Yes, the set of integers is closed under subtraction. This is because, for any two integers (say 3 & 5), their difference (in both directions) is an integer as well (i.e., both 3 - 5 and 5 - 3 are integers). Answer: Yes, it is closed.

Example 3: "The set of irrational numbers closed under addition". Provide an explanation supporting this statement. Solution: The sum of two irrational numbers is irrational always. i.e., we cannot find two irrational numbers whose sum is NOT irrational. Example: √2 + 2√2 = 3 √2, which is irrational. Answer: Supporting explanation is provided.

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