Question

The set of natural numbers is closed under the binary operation of
a. addition, subtraction, multiplication and division.
b. addition, subtraction, multiplication but not division.
c. addition and multiplication but not subtraction and division.
d. addition and subtraction but not multiplication and division.

Answer 1

Hint: Here the binary operation means the algebraic operation performed between any two numbers. And that operation may be, addition $\left( a+b \right)$, division $\left( a/b \right)$, subtraction $\left( a-b \right)$ and multiplication $\left( a\times b \right)$. If the obtained result after binary operation is a natural number, then it is considered in sets like, if (a) and (b) are natural numbers, then addition of $\left( a+b \right)$ is also a natural number.

Complete step-by-step answer:
It is given in the question that the set of natural numbers is closed under a binary operation. And we have been asked to find the binary operation from the options given. We know that natural numbers are counting numbers. The smallest natural number is 1.
Here, the binary operation refers to the algebraic operation performed between any two natural numbers. The performed operation may be addition $\left( a+b \right)$, division $\left( a/b \right)$, subtraction $\left( a-b \right)$ and multiplication $\left( a\times b \right)$.
In binary operation, we will have two inputs and we will get a single output. Here, we will select two natural numbers and on performing the algebraic operations, if we get the resulting number as a natural number, then we will consider it into our set.
If we add two natural numbers, a and b, we will get a natural number, c which is also a natural number. For example, (2 + 5) = 7. Here, both 2 and 5 are natural numbers and the result, 7 is also a natural number. Thus, addition operation is considered in our set.
Now, if we subtract two natural numbers, we may not get the resulting number as a natural number. For example, (5 - 7) = -2. Here, both 5 and 7 are natural numbers, but the resulting number, -2 is not a natural number. Thus, subtraction is not considered in our set.
If we divide two natural numbers, then it is possible that we may not get the resulting number as a natural number. For example, $\dfrac{15}{70}=\dfrac{3}{14}$. Here, both 15 and 70 are natural numbers but $\dfrac{3}{14}$ is not a natural number. Thus, division is not considered in our set.
Now, if we multiply two natural numbers, then we will get the resulting number as a natural number. For example, $15\times 3=45$. Here, both 15 and 3 are natural numbers and the resulting number, 45 is also a natural number. Thus, multiplication is considered in our set.
So, from the above observations, we can see that the binary operations possible for a set of natural numbers is addition and multiplication but not subtraction and division.
Thus, option (c) is the correct answer.

Note: Most of the students make mistakes while considering the case of subtraction. They subtract a big number from a small number and will get a natural number and in division they may divide a natural number with its factor, so the resulting number will be a natural number. And based on these observations, they might choose option (a) as the correct answer, but this is wrong. Subtraction and division are two such binary operations which do not result in closure of natural numbers all the time, so this must be kept in mind.