Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On R, defined * by a*b = a + 4b2
Here, Z + denotes the set of all non – negative integers.
Given that ‘*’ is an operation that is valid in the Real Numbers ‘R’ and it is defined as given:
⇒ a*b = a + 4b2, where a,b∈R,
Since a∈R and b∈R,
According to the problem it is given that on applying the operation ‘*’ for two given real numbers it gives a Real number as a result of the operation,
⇒ a*b∈R ...... (1)
Since b∈R then b2∈R,
We also know that the sum of two real numbers gives a real number. So,
⇒ a + 4b2∈R ...... (2)
From (1) and (2),
∴ The operation ‘*’ defines a binary operation on R.