# Let A=R*R and * be the binary operation on A defined by (a,b)*(c,d)=(a+c,b+d).Prove that * is both associative and commutative.Find the identity element for * on A.Also write the inverse element of the element (3,-5) in A.

(a,b)*(c,d) must be equal to (c.d)*(a,b)

LHS

As given in question

RHS

(c,d)*(a,b)

=> (c+a,d+b)=LHS

Hence Commutative

2. Associative

(a,b)*[(c,d)*(e,f)] must be equal to [(a,b)*(c,d)]*(e,f)

LHS

(a+c+e,b+d+f)

(Show steps)

RHS

Similarly

(a+c+e,b+d+f)=LHS

Hence Associative

3. Inverse element

a=3 and b=-5

to find (c,d) such that (a,b)*(c,d)=(1,1)

Therefore c=-2 and d=7.