Let * be a binary operation on `Q-{-1}` defined by `a`*`b=a+b+a b` for all `a ,\ b in Q-{-1}` . Then, Show that * is both commutative and associative on `Q-{-1}` . (ii) Find the identity element in `Q-{-1}`

Explanation

1.comutative means` a*b=b*a` , where` *` is binary operation.
`a*b=a+b+ab`(from defenation of binary operation)
`b*a=b+a+ba=a+b+ab`(since addition and mult. of nos is commutative)
therefore `a*b=b*a` i.e operation is commutative
2.associativity means `a*(b*c)=(a*b)*c`
`a*(b*c)=a+b+c+bc+ab+ac+abc`and `(a*b)*c=a+b+c+ab+bc+ac+abc`(since addition and multi of nos is commutative) therefore `a*(b*c)=(a*b)*c` i.e associative oroperty satisfied.
3.identity element means an element `x` such that `a*x=a`and `x*a=a`
now on solving `a*x=a` i.e `a+x+ax=a`on solving above eqn we get `x(a+1)=0`
`a+1` cant we 0 because `a` belongs to `Q-{-1}` if (a+1) is 0 then `a` is -1 which is not possible. therefore `x` is 0.
similary from `x*a=a` `x` is 0.i.e identity element is o.(zero).