Question

Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

Answer 1

a * b = `(("a" + "b")/2)`; a, b ∈ Q

For identity, a * e = e * a = a

Now; a * e = a

`("a"+ "e")/2` = a

a + e = 2a

e = 2a – a = a

Which is not possible

∴ Identity does not exist and hence the inverse does not exist.