Question

ABSTRACT MATH: How to show that an operation has no identity? Question shown as in textbook in the description (photo).

It has to do with the concept of algebraic identities, the odd symbol that looks like a multiply is just a binary operation.

Answer 1

See explanation...

Explanation:

I will use `ox` for the binary operation.

Given:

`a ox b = a^2b+ab^2`

Suppose `e in RR` is an identity for `ox` and `a != 0` is any non-zero real number.

Then:

`a = a ox e = a^2e+ae^2 = a(a+e)e`

Dividing both ends by `a` and transposing, we find:

`(a+e)e = 1`

Hence:

`a = 1/e-e`

But `a` is arbitrary, so this will not hold generally.

Hence we can deduce that there is no such identity `e`.