# What is closure property with examples

The closure property states that when elements of a set are performed on an arithmetic operation (addition (+), Subtraction(-) and multiplication(×) ) and the results are still a member of the given set then the set is close under that operation. Closure property applies to addition, Subtraction and multiplication. Let’s look at closure property in detail on each of the math operations.

## What is a closure property?

Closure property means when a set is performed on a particular operation and the results still belong to the given set, then the set is closed under that operation. For example, if two natural numbers are added together the result is still a natural number. Hence natural numbers are closed under addition.

## Closure property of addition

Closure property of addition states that when two or more numbers are added from a given set and the results still belong to that set, then that set is close under addition. Natural numbers, whole numbers, real numbers, integers, and rational numbers are all closed under addition. For example Natural numbers: if two natural numbers are added together the result is still a natural number. For example 3+4=7. 3 and 4 are natural numbers and the sum 5 is still a natural number.

## Closure property of multiplication

The closure property of multiplication states that when two numbers from a set are multiplied and the product still belongs to the set, then the set is closed under multiplication. Natural numbers, whole numbers, real numbers, integers, and rational numbers are all closed under multiplication. For example, if 2 and 3 are whole numbers, then 2×3=6. Since 2, 3, and 6 are whole numbers, then natural numbers are closed under multiplication.

## Closure property of Subtraction

In the closure property of Subtraction, only the sets of integers (Z) and rational numbers are closed under Subtraction. For example, if we subtract any two integers (Z) or rational numbers (Q) we get an integer or rational number.

Note: natural numbers and whole numbers are not closed under Subtraction.

Note also that no set of numbers can be closed under division if zero (0) is not a member of that set. Hence, Natural numbers (N), whole numbers (W), integers (Z), and rational numbers (Q) are not closed under division.