Question

Which of the following is not a binary operation on $R$ ?
A. $a \times b = ab$
B. $a \times b = a - b$
C. $a \times b = \sqrt {ab}$
D. $a \times b = \sqrt {{a^2} + {b^2}}$

Answer 1

In the given question, we are given four operations in the options and we are asked to find out which of them is not a binary operation. A binary operation on a set is an operation where two domains and the codomain are the same set. In other words, for $a \times b$ to be a binary operation on the set $R$, the resultant of the operation should also belong to the set $R$.

Complete step by step answer:
So, we will have to check each of the options one by one for the condition of binary operation so as to get to the right answer. Now, for $a \times b = ab$ on the set $R$, the numbers a and b belong to the real number set. The resultant of the operation $a \times b = ab$ is the product of the two numbers. We know that the product of two real numbers is always a real number. Hence, the resultant of the operation also belongs to the set $R$.Hence, the operation $a \times b = ab$ is a binary operation on $R$.

Similarly, for $a \times b = a - b$ on set $R$, the resultant of the operation is the difference of the two numbers. We know that the difference of two real numbers is always a real number. Hence, the resultant of the operation also belongs to the set $R$. Hence, the operation $a \times b = a - b$ is a binary operation on $R$. Now, for $a \times b = \sqrt {{a^2} + {b^2}}$, the result of the operation is the square root of the sum of squares of the numbers.

Now, we know that the sum of squares of real numbers is always a positive real number. Also, the square root of a positive real number is always a real number. Hence, the resultant of the operation also belongs to the set $R$.Hence, the operation $a \times b = \sqrt {{a^2} + {b^2}}$ is a binary operation on set $R$. Now, for $a \times b = \sqrt {ab}$, the numbers a and b belong to the real number set. The resultant expression of the operation is the square root of the product of the two numbers.

We know that the product of two real numbers is a real number. But the square root of a real number is not always a real number if the real number inside the square root is negative.
If the real number inside the square root is negative, then the square root is a complex number. So, the resultant of the operation does not always belong to the set $R$. Hence, the operation $a \times b = \sqrt {ab}$ is not a binary operation on the set $R$.

Hence, option C is correct.

Note: One must know the definition of a binary operation to solve such a question.We must also have an idea about the calculation of the resultant of any operation.Addition, subtraction and multiplication are binary relations on the real number set. This property can be used directly in such questions to save time.