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Closure property states that when we add or multiply numbers of a given set, resulting number should also be from that set only. If the result obtained after multiplication or the addition doesn't belong to the set then it means that the set does not follow closure property. It is the closed set satisfying the closure property.

If a set has the closure property under a given operation, then we say that the set is closed under the operation. Closure Property demonstrates whether the system of numbers is closed or not under different mathematical operations. It is easier to understand a property by looking at examples than it is by simply talking about it in an abstract way. So given below are few examples.

  1. 5 $\times$ 4 = 20 is closed under multiplication as 5, 4, 20 are real numbers.
  2. The set of odd natural numbers, 1, 3, 5, 7,….., is closed by the addition because the sum of any two of them is another odd number.
  3. 10 - 20 = -10. Here 10 and 20 are whole numbers and -10 is not. Difference of two numbers yields a result that is not a whole number. The given set of whole numbers does not possess the closure property for subtraction.

Closure Property of Addition

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Closure property of real number addition states that the sum of any two real numbers equals another real number.
Let x, y be two integers. If x + y is also an integer, then the set of integers is said to be closed under the operation of addition.
For any two elements x, y $\epsilon$ Z , x + y $\epsilon $ Z is the closure property of addition.The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers. 

Given below are some examples for closer property under addition

5 + 4 = 9 as 5 + 4 = 9 $\epsilon$ Z

-1 + (-2) = -3  $\epsilon$ Z

3 + (-5)  = -2 $\epsilon$ Z

8 + 4 = 12 $\epsilon$ Z

Closure Property of Subtraction

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Real numbers holds closure property of subtraction like closure property of addition. But it is not true for the whole numbers. In closure property of subtraction the difference of any two whole numbers will not always be a whole number. That is, system of whole numbers is not closed under subtraction.

Examples: 5 - 2 = 3

9 - 9 = 0

5 - 15 = - 10

5 - 9 = -4
From the above first and second example we see that the difference of two numbers is always a whole number. But from examples, 3 and 4, we notice that the the difference of whole numbers is not a whole number, the result obtained from 3 and 4 examples is an integer.

Closure Property of Multiplication

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Closure property of real number multiplication states that the product of any two real numbers equals another real number. The product of two numbers doesn't change even if the order of the numbers is changed.

Let x, y be two integers. If x $\times$ y is also an integer, then the set of integers is said to be closed under the operation of multiplication.

Given below are some examples for closer property under multiplication.
For any two elements x, y $\epsilon$ Z , x $\times$ y $\epsilon $ Z is the closure property of multiplication.

Example: 4 $\times$ 3 = 12

12 $\times$ 6 = 72

9 $\times$ 12 = 108

When two elements in the set are combined the result is also included in the set. In the above example observe that the factors and products are also real numbers.

The closure property of real number multiplication states that when we multiply real numbers with other real numbers the result is also real.

Closure Property of Division

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The closure property of division same as subtraction under closer property, system of whole numbers is not closed under division. That is, division of any two whole numbers is not always a whole number.

Examples:

$\frac{1}{2}$ = 0.5

$\frac{4}{12}$  = 0.33

$\frac{1}{4}$ = 0.25

$\frac{27}{9}$ = 3

From the above first three examples we see that the division of two numbers is not always a whole number. The result obtained is a fraction. Last result obtained is a whole number. We can say that division of two whole numbers is not always a whole number.