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.
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.
Closure property is mainly divided into 4 parts:
Closure Property of Addition
The closure property of addition states that the sum of any two elements of a given set always belongs to the same set.
Examples:
- For whole numbers, if a and b are whole numbers, then a + b is also a whole number.
- For integers, the sum of any two integers is always an integer.
Take another 2 even numbers, 4 + 6 = 10. The result is also an even number. Hence, the closure property under addition is satisfied.
| Closure Property of Addition | |
|---|---|
| Real Numbers | a + b = Real number (a, b are real numbers.) |
| Rational Numbers | a + b = Rational number (a, b are Rational 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 Rational Numbers.) |
| Integers Numbers | a - b = Integer (a, b are integers.) |
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 and also belongs to the given set, so the closure property is satisfied in this case.
But if we multiply 8 × 2 = 16, the result is even but does not belong to the given set, so the set {2, 4, 8} is not closed under multiplication
| Closure Property of Multiplication | |
|---|---|
| Real Numbers | a × b = Real number (a, b are real numbers.) |
| Rational Numbers | a × b = Rational number (a, b are Rational 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 not in the original number set. Therefore, the closure property is not satisfied under division for this set because the result does not belong to the same set.
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.
Related Articles:
Closure Property Solved 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:
No, it is not closed. Since 6 × 18 = 108 and 108 is not in the set, closure fails.
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).