Let * be a binary operation on set `Q-[1]`defined by `a*b=a+b-a b`for all`a , b in Q-[1]dot`Find the identity element with respect to `*onQdot`Also, prove that every element of `Q-[1]`is invertible.

Let * be a binary operation on set `Q-[1]`defined by `a*b=a+b-a b`for

1.	Let * be a binary operation on set `Q-[1]`defined by `ab=a+b-a b`for all`a , b in Q-[1]dot`Find the identity element with respect to `onQdot`Also, prove that every element of `Q-[1]`is invertible.
Answer» Let `e` is the identity element for the given operation. Then, `ae = a` `=>a+e-ae = a` `=> e(1-a) = 0` `=> e = 0 and a = 1` It is given that `Q` does not contain `1`. `:. a` can not be `1`. `:. e = 0` So, identity element for the given operation is `0`. Now, we will find the invertible elements of `Q-[1]`. Let `x` is the inverse of `a` Then, `ax = e` `=>a+x-ax = 0` `=>x(1-a) = -a` `=>x = a/(a-1)` So, we will not be able to find the inverse if `a = 0`. For, all other values of `a`, we can find the inverse. So, all elements of `Q-[1]` are invertible.