Question

Closure property is observed w.r.t _______ operation in integers.
\[\begin{align}
  & A.+,\times \\
 & B.+,\div ,\times \\
 & C.+,\times ,- \\
 & D.+,-,\div \\
\end{align}\]

Answer 1

Hint:
 In this question, we need to find an operation for which closure property holds on integer. For this we will first understand the meaning of closure property and after that we will check it for all four operations that are addition (+), subtraction (-), multiplication (×) and division (÷).

Complete step by step answer:
Let us first understand the meaning of closure property. According to the closure property, any two elements in a set combines to produce a resultant element in the same set.
Here we need to check that, when we apply any of four options on integers the result will be integer too.
Let us check for addition (+), subtraction (-), multiplication (×) and division (÷).
For addition:
When two integers are added, we get the answer as integer only. For example: 2+23 = 25 and -2+5 = 3. Therefore, closure property is observed with respect to addition (+) property.
For subtraction:
When two integers are subtracted, we get the answer as integer only. For example: 2-3 = -1 and 97-23 = 74. Therefore, closure property is observed with respect to subtraction (-) property.
For multiplication:
When two integers are multiplied, we get the answer as an integer only. For example: $ 5\times 9=45\text{ and }-4\times 6=-24 $ . Therefore, closure property is observed with respect to multiplication (×) property.
For division:
When two integers are divided, we may or may not get the answer as an integer. For example: $ 2\div 3=\dfrac{2}{3} $ is not a integer but $ 4\div 2=\dfrac{4}{2}=2 $ is a integer. Therefore, closure property is not observed with respect to division (÷) property.
Hence we can conclude that closure property is observed with respect to addition (+), subtraction (-) and multiplication (×) operation.
Hence option C is the correct answer.

Note:
Students should carefully check for all properties keeping in mind the definition of integers. Note that subtraction is not closed in the set of natural numbers. Similarly, we can check closure property for any set.