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.

Let A = N xx N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative.

Let A = N×N 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, if any.

Let A" "=" "NxxN 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, if any.

Let * be a binary operation on Z defined by a*b= a+b-4 for all a ,\ b in Zdot Show that * is both commutative and associative.

Let A=NxNa n d^(prime)*' be a binaryoperation on A defined by (a , b)*(C , d)=(a c , b d) for all a , b , c , d , in Ndot Show that '*' is commutative and associative binary operation on A.

Let A=NxNa n d^(prime)*' be a binaryoperation on A defined by (a , b)*(C , d)=(a c , b d) for all a , b , c , d , in Ndot Show that '*' is commutative and associative binary operation on A.

Let A = Q xxQ and ** be a binary operation on A defined by (a, b) ** (c,d) = (ad + b, ac). Prove that ** is closed on A = QxxQ . Find (i) Identity element of (A, **) , (ii) The invertible element of (A,**).

Let R_0 denote the set of all non-zero real numbers and let A=R_0xxR_0 . If * is a binary operation on A defined by (a ,\ b)*(c ,\ d)=(a c ,\ b d) for all (a ,\ b),\ (c ,\ d) in Adot Show that * is both commutative and associative on A (ii) Find the identity element in A

Let A=Q x Q and let * be a binary operation on A defined by (a , b)*(c , d)=(a c , b+a d) for (a , b),(c , d) in Adot Then, with respect to * on A Find the identity element in A Find the invertible elements of A.

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.