Question

A binary operation *,defined on the set of R of real numbers is given by x*y=x+y+2xy. Find the identity element? Show workings please

Answer 1

Identity element `e=0`.

Explanation:

The identity element `e` of a set `S` equipped with an operator `*` is defined such that `x*e=x` and `e*x=x` for `AAx inS`.

Let `e` be the identity element in `RR` equipped with group operator `*` defined as `x*y=x+y+2xy`. We know that `x*e=x`, so `x+e+2xe=x`. Note that `e*x=x*e` because the group operator `*` is commutative so that we only need to consider the equation `x+e+2xe=x`.
`e+2xe=0`
`e(1+2x)=0`
`:.e=0`
Note that `x=-1/2` is also a solution of `e(1+2x)=0`, but `-1/2` is the zero element (`-1/2*x=-1/2` and `x*-1/2=-1/2`)