Question

If you define the binary operation `x "*" y = x+y-xy` for any `x, y in ZZ` then what is the identity?

Answer 1

`0`

Explanation:

I don't know if you wanted to show that `"*"` is commutative, but note that for any `x, y in ZZ` we find:

`x"*"y = x+y-xy = y+x-yx = y"*"x`

Note that for any integer `x`:

`x"*"0 = x+0-x(0) = x`

`0"*"x = 0+x-0(x) = x`

So `0` is the (two-sided) identity.

Footnote

Note that the two-sided identity of a binary operation is necessarily unique:

Suppose `e_1` and `e_2` are identities of some binary operation `@`.

Then:

`e_1 = e_1 @ e_2 = e_2`