Closure Property

According to the Closure Property, if an arithmetic operation (such as addition, subtraction, multiplication, or division) on any two numbers in a set always gives another number from the same set, then the set is considered to be closed under that operation. 

For example: 

Since the sum of any two natural numbers is always another natural number, the set of natural numbers is closed under addition.

For instance, 2 + 4 = 6, which is also a natural number

We know that subtracting a greater number from a smaller one can result in a number that is not in the set. Therefore, the set of natural numbers is not closed under subtraction.

For instance, 6 subtracted from 3 is -3 (not a natural number).

The formula for the closure property is:

∀ a, b ∈ S ⇒ a (operation) b ∈ S (operations can be +, -, ×, and ÷)

Where:

S: a given set.

Operator: any mathematical operation, including addition, subtraction, multiplication, and division.

All fundamental arithmetic operations close the set of real numbers except dividing any number by zero. For any two real numbers a and b, the closure property can be expressed as given below:
 

A set of real numbers is made up of both rational and irrational numbers. 

Real numbers consist of:

• Natural numbers: {1, 2, 3, …}
• Whole numbers: {0, 1, 2, 3,...}
• Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
• Rational numbers
• Irrational numbers

According to the Closure Property, real numbers are closed under addition, subtraction, multiplication, and division (except division by zero, which is undefined).

Closure Property for Integers

The representation of the set of numbers is Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

The closure property applies to the arithmetic operations such as addition, subtraction, and multiplication of integers, but not to division.

 Closure property under Addition

According to this property, the sum of any two integers always results in another integer. That is, for the integers a and b, their sum (a + b) is also an integer.
For instance:

 (–7) + 9 = 2.

 5 + 12 = 17.

 Closure property under Subtraction

The difference between two integers will always result in an integer. That is, for the integers a and b, (a – b) will also be an integer.

 For instance:

 12 – 7 = 5.

 (– 8) – (– 2) = – 6

 Property of closure under Multiplication

When two integers are multiplied, their product will always be an integer. For the integers a and b, their product (a × b)  will also be an integer.

 For instance:

 4 × (-6) = -24

 (-9) × (-5) = 45

 Property of closure under Division

Division does not always yield an integer. So, integers are not closed under division. 
 
 For instance:

 (-15) ÷ 3 = -5 (an integer) 

(-8) ÷ (-20) = 0.4 (not an integer)
 

Numbers that may be represented as p/q, where p and q are integers and q ≠ 0, are considered rational.
Rational numbers are closed under addition, subtraction, and multiplication. Since division by zero is undefined, the closure property does not apply to division when the divisor is zero. 
Since division by zero is undefined, rational numbers are not closed under division when the divisor is zero. 
For instance:
5/9 + 3/4 = 41/36 (a rational number)

7/8 – 1/6 = (21 - 4)/24 =  40/48 (a rational number)

2/7 × 5/6 = 10/42 (a rational number)
 

The set of whole numbers is represented as W = {0, 1, 2, 3, 4, …}, which contains all numbers, including 0.
Whole numbers are closed under addition and multiplication. This implies that for whole numbers a and b, the result of a + b and a × b will also be whole numbers.
For instance:
10 = 3 + 7.
4 × 5 = 20 (whole number)

Subtraction and division are not closed for whole numbers.  Accordingly, a – b or a ÷ b could not always provide a whole integer.
For instance:

9 – 4 = 5 (whole number)

4 – 9 = –5 (Not a whole number)

There is a whole number: 12 ÷ 3 = 4.

1 ÷ 6 equals 0.166... (not a whole number, but a decimal).
 

If any two numbers from a set add up to a number that belongs to the same set, the set is considered closed under addition.
Addition is closed for all integers, rational numbers, whole numbers, natural numbers, and real numbers.
 

If the product of two numbers from a set results in a number within the same set, we can say that the set is closed under multiplication.
 

When subtracting two numbers from a given set, and the result belongs to the same set, we confirm that the given set is closed under subtraction. This property holds for real numbers, rational numbers, and integers.
 

What is the Closure Property of Division?
 

When two numbers are divided, and the resulting value belongs to the same set, the set is said to be closed under division. However, this property does not apply to most number sets. The closure property of division does not satisfy integers and whole numbers. 

The closure property lays the foundation for various mathematical concepts like algebra. The applications of the closure property are widespread across various fields beyond mathematics. Let’s now look at its different applications:

• Financial transactions: In financial transactions like withdrawing or depositing money from a bank, real numbers are used to represent the total balance

• Shopping and Discounts: When purchasing multiple products, we calculate the discounts or total cost using the closure property of addition.

• Time calculations: Adding or subtracting time always yields a valid time.
  For example: If the total duration of an exam is 3 hours and the cool-off time is 20 minutes, then the total time for the exam is: 3 hours + 20 minutes = 3 hours 20 minutes, which remains a valid time.