[Maths Class Notes] on Closure Pdf for Exam

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.