Question

Which of the following statements is incorrect?
A) Whole numbers are closed under addition.
B) Whole numbers are closed under multiplication.
C) Whole numbers are closed under subtraction.
D) Whole numbers are not closed under subtraction.

Answer 1

Hint: We will first understand the meaning of being closed under any operation. Then, after that, we will inspect all options one by one to verify which of the statements is correct and which among them is incorrect.

Complete step-by-step answer:
Let us understand what we mean by “closed”.
We see that any set X is closed under a binary operation ‘*’ if $\forall a,b \in X$, we have $a*b \in X$.
In other words, if the binary operation keeps the set intact and gives the result of any two numbers from the set in the set only, then the set X is said to be closed under ‘*’.
Let us go through all the options one by one now:
Option A: Whole numbers are closed under addition
We see that if we have a and b from the set of whole numbers, then a + b will always be a whole number. For example: 2 and 5 are whole numbers and 2 + 5 = 7 is a whole number as well, similarly 0 and 8 are whole numbers and 0 + 8 = 8 again a whole number.
Hence, option (A) is correct.
Option B: Whole numbers are closed under multiplication
We see that if we have a and b from the set of whole numbers, then $a \times b$ will always be a whole number. For example: 2 and 5 are whole numbers and $2 \times 5 = 10$ is a whole number as well, similarly 0 and 8 are whole numbers and $0 \times 8 = 0$ again a whole number.
Hence, option (B) is correct.
Option C: Whole numbers are closed under subtraction
We see that if Whole numbers are closed under subtraction. Then for any a and b from the set of “Whole Numbers”, their subtraction must lead to a whole number as well. So, take a = 0 and b = 5. Then, a – b = 0 – 5 = -5 which is not a whole number.
Hence, option (C) is incorrect.
Since, option (D) is opposite of option (C) and option (C) is incorrect, therefore, (D) must be correct.

Hence, the required answer is (C).

Note: The students must know that the set of whole numbers is defined as: {0, 1, 2, 3, …….}.
It is basically the set of Natural Numbers with 0 added to it as well.
The students must note that, if we need to prove something to be correct, we cannot give an example to do so. But to prove something incorrect, we just need to give one example which does not satisfy it. Like in option (C), if we would have taken a = 5 and b = 0 then a – b would have been a whole number. Therefore, we took a contradictory example.