Question

What is closure property?

Answer 1

Hint: Closure Property basically states that, if in two operands an operation is applied in which both the operands belong to the same set, then the value after that operation, the result should belong to the same set only in which the operands belong. For example, if a and b belong to real numbers, then after any operation like addition, multiplication or subtraction, the output should be in the set of real numbers only.

Complete step-by-step answer:
According to closure property, if two numbers belong to the same set and an operation is performed between them, then the result should be in the same set only.
Since, we know that we can perform four main operations that is addition, subtraction, multiplication and division.
Let’s check which operation does this property holds:
We are taking two values from the set of real numbers, suppose we take $3.2,1.5$.

1.Addition: Adding the two values above and we get:
$3.2 + 1.5 = 4.7$
We can see that the result obtained belongs to a real number set, which proves that in addition two real numbers give a real number only.
Therefore, the real numbers are closed under addition.

2. Subtraction: Subtracting the two values above and we get:
$3.2 - 1.5 = 1.7$
Again, we can see that the result obtained belongs to the real number, which proves that subtraction of two real numbers gives a real number only.
Therefore, the real numbers are closed under subtraction.

3.Multiplication: Multiplying the two values $3,6$ which are real numbers and we get:
$3 \times 6 = 18$
We can see that the result obtained belongs to the real number, which proves that multiplication of two real numbers gives a real number only.
Therefore, the real numbers are closed under multiplication also.

4.But for division, closure property would not work, because suppose we divide any real number with zero it would give an undefined value which would not belong to the real number set. For example, we take $2,0$ on dividing them we get:
$\dfrac{2}{0} = undefined$ and the set is broken.

Overall, we can say that addition, multiplication and subtraction for real numbers works on closure property. But, division does not.

Note: Similarly, we can check for integers, rational numbers and etc.
If any single value returned after the operation is performed does not belong to the set of the operands, it will not hold the closure property.
For two operands, we can also make a binary table to check for closure property.