Dhristi JEE 2022-24

Let’s Understand the Basics of Closure Property

Closure Property  Definition (Maths)

In mathematics, Closure refers to the likelihood of an operation on elements of a set. If something is closed, then it means if an operation is conducted on any of the two elements of the set, then the result of that operation is also within the set. If the elements in a set exist for which the result of the operation is not within the set then the operation will not be termed as closed.


Examples of Closure Property

  • Taking into account the addition of the natural numbers where the natural numbers are set and the process of addition is the operation. When a natural number is added to a natural number, it will always result in a natural number, and then it is termed that the addition is closed on natural numbers. 

  • Now considering the division of the natural numbers, we can observe that for two numbers 2 and 3, the result of 2 divided by 3 will be ⅔. As ⅔ is not the natural number, the division on the natural number is not closed.


Closure for Different Functions 

Closure for Multiplication: The elements of real numbers in a set are closed under multiplication. If you do the multiplication of two real numbers, you get another real number. There is no probability of ever getting anything other than the real number. 


For Example:

4*5 = 20 


3½ * 2½ = 8 ¾ 


1.5  * 2.1 = 3.15


Closure for Addition: The elements of real numbers in a set are closed under addition. The addition of the two real numbers gives another real number. There is no probability of ever not getting anything other than the real number. 


For Example:

5 + 12 =17


3½ + 6 = 9½ 


Conclusion

Whenever we use the term "closure" in mathematics, it is applicable to sets and mathematical operations. The sets can include basic numbers, vectors, matrices, algebra, etc. The operations can include any mathematical operation like addition, multiplication, square root, etc.

FAQs on Closure

1. Is the set of irrational numbers considered to be closed under the algebraic operations?

No, for the set of numbers to be closed under the assigned operation, any pair of the numbers with that operation will result in the number, within the set or in reverse. If you find the pair of numbers in the set, where the operation results in the number which is not in the set then that specific set is not considered closed over the specific operation. 

2. What is the meaning of closed under addition?

For any two members of the set, the addition will return a value that is also a member of the set given that the set is closed. The addition of a set of positive integers is closed because the sum of two integers is an integer. The addition of 2 digit integers is not closed for various reasons. For instance 60 + 62 = 122, which is the 3 digit number. The issue is because the original set was 2 digit numbers.     

3. What are closed numbers?

In maths, the natural numbers are closed under multiplication and addition. The set is said to be closed only if the operation on two elements within the set gives another element of a similar set. If the operation gives even one element outside the set, that operation does not give closure. A set is said to be closed under the operation if an operation on these set members always tends to give a member of that specific set. 

4. What do you mean by closure property?

A set is said to be closed for any mathematical operation if it has closure property. Simply a set is said to be closed under an operation if conducting that operation on members of the set always yields a member of that set. For example, the positive integers are not closed under subtraction, but are under addition: 1 − 2 is not a positive integer despite both 1 and 2 are positive integers.

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