Solution:
We are given three binary operations defined on the set Q of rational numbers.We need to find the identity element in each case.
Let e denote the identity element which is also a rational number.
Let’s recall the definition of an identity element.
For any element x belong to a set with binary operation ∗, if e denotes the identity element, then we have x∗ e = x = e ∗ x.
Consider the binary operation a ∗ b = a2+b2.
Let’s compute a ∗ e.
a ∗ e = a2 + e2
Now, we know that + is a binary operation on Q and a is an element of Q. This implies that a2∈Q.
But what we need is a ∗ e = a = e ∗ a for every a ∈ Q.
That is, we need a2+e2 = a for every a.
This is not possible.
Therefore, there does not exist an identity element for the binary operation a ∗ b = a2 + b2.
Hence, the given statement is false.
