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.

1. Commutative
(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.

 
  • 5
Hu
  • 0
Where is inveese element
  • -1
(a,b)+(c,d)=(a+c,b+d)
  • 0
Such that (a,b)+(c,d)=(a+c,b+d)
  • 0
What are you looking for?