On the set `R-{-1}` a binary operation `*` is defined by `a*b=a+b+a b` for all `a , b in R-1{-1}` . Prove that * is commutative as well as associative on `R-{-1}dot` Find the identity element and prove that every element of `R-{-1}` is invertible.

On the set Z of integers a binary operation * is defined by a*b= a b+1 for all a ,\ b in Z . Prove that * is not associative on Zdot

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}

On R-[1] , a binary operation * is defined by a*b=a+b-a b . Prove that * is commutative and associative. Find the identity element for * on R-[1]dot Also, prove that every element of R-[1] is invertible.

Show that the binary operation * on A=R-{-1} defined as a*b=a+b+a b for all a ,bA is commutative and associative on Adot Also find the identity element of * in A and prove that every element of A is invertible.

Show that the binary operation * on A=R-{-1} defined as a*b=a+b+a b for all a ,bA is commutative and associative on Adot Also find the identity element of * in A and prove that every element of A is invertible.

Let * be a binary operation on Q_0 (set of non-zero rational numbers) defined by a*b= (3a b)/5 for all a ,\ b in Q_0 . Show that * is commutative as well as associative. Also, find the identity element, if it exists.

For the binary operation * defined on R-{1} by the rule a * b=a+b+a b for all a ,\ b in R-{1} , the inverse of a is

Discuss the commutativity of the binary operation * on R defined by a*b=a^2 b for all a,b in Rdot

Let * be a binary operation on the set Q_0 of all non-zero rational numbers defined by a*b= (a b)/2 , for all a ,\ b in Q_0 . Show that (i) * is both commutative and associative (ii) Find the identity element in Q_0 (iii) Find the invertible elements of Q_0 .

Let A = R × R and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative and associative. Find the identity element for * on A.