Closure Property – Definition, Formula, Addition, Multiplication, Division

This idea is important when working with number systems such as natural numbers, whole numbers, integers, rational numbers, and real numbers. Here, we explain how the Closure Property applies to arithmetic operations, with examples that help show when the property is followed and when it is not.

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What is Closure Property?

The Closure Property refers to a rule in mathematics which states:

When an arithmetic operation is performed on any two elements of a set, and the result always belongs to the same set, then the set is said to be closed under that operation.

For example, consider a set of integers. If we take two integers a and b, their:

• Sum (a + b),

• Difference (a − b), and

• Product (a × b)

will always be integers. Hence, integers are closed under addition, subtraction, and multiplication. 

However, division is different. For example, 3 ÷ 2 = 1.5. Since one point five is not an integer, this means that integers are not closed under division.

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Closure Property of Real Numbers

Real numbers are closed under arithmetic operations like addition, subtraction, and multiplication. They are also closed under division, except when the divisor is zero.

Examples:

• 2 plus 3 equals 5 (real number)

• 5 minus 1 equals 4 (real number)

• 4 times 2 equals 8 (real number)

• 6 divided by 3 equals 2 (real number)

• 7 divided by 0 is undefined (not closed)

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Closure Property of Rational Numbers

Rational numbers include fractions and terminating or repeating decimals. They are closed under all basic operations except division by zero.

Examples:

• One-half added to one-third equals five-sixths( rational number)

• Three fourths minus one half equals one fourth (rational number)

• One half times two thirds equals one third (rational number)

• Three fourths divided by one half equals three halves (rational number)

• Any number divided by zero is undefined (not closed)

Closure Property of Integers

Integers are closed under addition, subtraction, and multiplication. Integers are not closed under division because dividing two integers can result in a value that is not an integer.

Examples:

• Negative four plus six equals two (integer)

• Ten minus thirteen equals negative three (integer)

• Negative three times five equals negative fifteen (integer)

• Eight divided by three equals two point six seven (not an integer)

Closure Property in Modular Arithmetic 

In modular arithmetic, operations like addition, subtraction, and multiplication are closed because results always stay within the range 0 to (modulus − 1).

Examples (mod 5):

• 2 + 4 = 6 → 6 mod 5 = 1 (in the set {0, 1, 2, 3, 4})

• 3 × 2 = 6 → 6 mod 5 = 1 (in the set

•  4 − 1 = 3 → 3 mod 5 = 3 (in the set)

•  1 − 3 = −2 → (−2 mod 5 = 3) (in the set)

So, modular arithmetic is closed under these operations.

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Closure Property Types

The closure property can be checked using the four basic arithmetic operations: 

• Addition

• Subtraction

• Multiplication

• Division

Let’s explore how closure applies to each operation across various number systems.

Closure Property of Addition

A set is closed under addition when the sum of any two numbers from that set is still part of the set. In simple terms, if you add two numbers from the set, the result will also belong to the same set..

For example:

• Natural Numbers (N): 4 + 2 = 6 → 5 ∈ N (Closed)

• Integers (Z): -1 + 6 = 5 → 5 ∈ Z (Closed)

• Rational Numbers (Q): ½ + ¾ = 5⁄4 ∈ Q) (Closed)

• Real Numbers (R): π + 2 ∈ R (Closed)

• Irrational Numbers: √2 + (−√2) = 0 ∉ irrational (Not Closed)

Note: Non-zero integers are not closed under addition because −1 + 1 = 0, and 0 is not part of that set.

Closure Property of Subtraction

The closure property of subtraction means that a set is closed under subtraction if, when you subtract any two elements from the set, the result is also part of the set.

Examples:

• Integers (Z): 4 − 8 = −4 → −4 ∈ Z (Closed)

• Rational Numbers (Q): ¾ − ½ = ¼ ∈ Q (Closed)

• Real Numbers (R): π − 2 ∈ R (Closed)

However, some sets are not closed under subtraction:

• Natural Numbers (N): 3 − 5 = −2 → −2 ∉ N (Not Closed)

• Whole Numbers (W): 0 − 3 = −3 ∉ W (Not Closed)

Note: For a set to be closed under subtraction, every possible difference of its elements must still belong to the set.

Closure Property of Multiplication 

The closure property of multiplication means that a set is closed under multiplication if the product of any two elements from the set is also an element of the set.

Examples:

• Natural Numbers (N): 5 × 6 = 30 ∈ N → Closed

• Whole Numbers (W): 0 × 7 = 0 ∈ W → Closed

• Integers (Z): −9 × 6 = −54 ∈ Z→ Closed

• Rational Numbers (Q): ²⁄₃ × ¾ = ⁶⁄₁₂ ∈ Q → Closed

Exception:
The set of irrational numbers is not closed under multiplication. For example,  √2 × √8 = √16 = 4, which is not irrational. To satisfy closure under multiplication, every product of two elements in the set must also belong to the set.

Closure Property of Division

The closure property of division means that a set is considered closed under division if dividing any two of its elements (except when dividing by zero) always results in an element within the same set.

Examples:

• Rational Numbers (Q): ½ ÷ ⅓ = ³⁄₂ ∈ Q → Closed (if not dividing by 0)

• Real Numbers (R): 10 ÷ 2 = 5 ∈ R → Closed (except division by 0)

Not Closed For:

• Integers (Z): 7 ÷ 2 = 3.5 ∉ Z

• Whole Numbers (W): 6 ÷ 4 = 1.5 ∉ W

Note: Division by zero is undefined in all number systems, so sets are never closed under division when zero is the divisor.

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Closure Property Formula

The Closure Property Formula describes a rule in mathematics that helps determine whether a set is closed under a specific operation such as addition, subtraction, multiplication, or division.

Let a and b be any two elements of a set S, and let ∘ represent an arithmetic operation (such as +, −, ×, ÷). Then:

∀a,b ∈ S ⇒ a ∘ b ∈ S

Here, 

• "∀" (for all): This means the rule applies to every possible pair of elements in the set.

• "a, b ∈ S": Both a and b are elements (members) of the set S.

• "a ∘ b ∈ S": After performing the operation (addition, subtraction, etc.) on a and b, the result must also be in the same set S.

Closure Property Examples

Now that we have understood the concept of the Closure Property and how it applies to different arithmetic operations (addition, subtraction, multiplication, and division), let’s explore a few solved examples. 

Closure Property Example 1: Is the set {0, 2, 4, 6, 8} closed under addition?

Solution: We will test a few additions using elements from the set:

• 2 + 4 = 6 → 6 is in the set

• 0 + 8 = 8 → 8 is in the set

• 4 + 6 = 10 → 10 is not in the set

Since 10 is not in the set, the set fails closure under addition.

Closure Property Example 2: Is the set {0, 1, 2, 3} closed under subtraction?

Solution: Let’s test some subtraction operations:

• 3 − 2 = 1 → 1 is in the set

• 2 − 0 = 2 → 2 is in the set

• 1 − 3 = −2 → −2 is not in the set

Since −2 is not in the set, the set is not closed under subtraction.

Closure Property Example 3: Is the set {1, 3, 9, 27} closed under multiplication?

Solution: Check a few products:

• 1 × 3 = 3 → in the set

• 3 × 9 = 27 → in the set

• 9 × 27 = 243 → 243 is not in the set

Since 243 is not in the set, it violates the closure condition.

Closure Property Example 4: Is the set {1, 2, 4, 8} closed under division?

Solution: Try dividing elements from the set:

• 8 ÷ 4 = 2 → 2 is in the set

• 4 ÷ 2 = 2 → 2 is in the set

• 2 ÷ 4 = 0.5 → 0.5 is not in the set

Since 0.5 is not included in the set, the set is not closed under division.

Closure Property Example 5: Is the set of integers Z closed under addition?

Solution: Test any pair of integers:

• −2 + 5 = 3 → 3 is an integer

• 0 + (−4) = −4 → also an integer

Since the sum of any two integers is always an integer, the set of integers is closed under addition.

Also Read: Euler's Formula

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