Question

Closure property is not applicable to the .................. operation of integers.
(a) addition
(b) subtraction
(c) multiplication
(d) division

Answer 1

Hint: For answering this question first we will see what closure property is and how it is applicable in various arithmetic operations of integers. After that, we will take up some examples for better understanding and answer the question correctly.

Complete step-by-step answer:
Given:
We have to answer that closure property is not applicable in which operation of integers.
Now, before we proceed we should know what closure property is.
Closure Property:
Among the various properties of integers, closure property is one of the fundamental properties of integers. According to it, if $a$ and $b$ are any two integers and if we do an arithmetic operation between $a$ and $b$ to get a new result $c$. And if $c$ is also an integer then, closure property is valid. We have the following two types of closure properties:
1. Closure property under addition and subtraction states that the sum or difference of any two integers will always be an integer i.e. if $a$ and $b$ are any two integers then $a+b$ and $a-b$ will also be an integer. For example: 3 – 4 = 3 + (-4) = -1 , which are integers.
2. Closure property under multiplication states that the product of any two integers will be an integer i.e. if $a$ and $b$ are any two integers then $ab$ will also be an integer. For example: $5\times 2=10$, $6\times 3=18$, which are integers.
Now, though the closure property is valid for the case of addition, subtraction and multiplication but the division of integers doesn’t follow the closure property, i.e. the quotient of any two integers $a$ and $b$, may not be an integer always. For example: $5÷10=0.5$ , is not an integer.
Thus, we conclude that closure property is not valid for the division of integers.
Hence, (d) is the correct option.

Note: Here, the students should first understand what is asked in the question and then proceed in the right direction to get the correct answer. After that, we should try to understand closure property with the help of examples, and this will strengthen our basics of the number system.