check whether the operation * defined on the set A=RxR or (a,b) *(c,d)= (a+c, b+d) is a binary operation or - Maths -

Question

check whether the operation * defined on the set A=RxR or (a,b)
*(c,d)= (a+c, b+d) is a binary operation or not if it is a binary
operation is it commutative and associative too?​

Rishabh Mittal · Accepted Answer

Dear Student ,

Please find below the solution to the asked query :

Closure law:Let a,b , c,d ∈ R×R be any two elements
Then, a,b*c,d=a+c ,b+d  ∈ R×RTherefore , * is a binary operation on R x R 
Commutativity:Let a,b , c,d ∈ R×R be any two elements
⇒a,b*c,d=a+c ,b+d =c+a , d+b=c,d * a,bSo, * is commutative on R × RAssosciativity:Let a,b , c,d, e,f ∈ R×R be any three elements
⇒a,b*c,d*e,f=a+c ,b+d *e,f=a+c+e , b+d+f⇒a,b*c,d*e,f=a,b*c+e , d+f=a+c+e , b+d+fSo, a,b*c,d*e,f=a,b*c,d*e,fThus, * is assosciative on R × R
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check whether the operation * defined on the set A=RxR or (a,b) *(c,d)= (a+c, b+d) is a binary operation or not . if it is a binary operation is it commutative and associative too?​

check whether the operation * defined on the set A=RxR or (a,b) *(c,d)= (a+c, b+d) is a binary operation or not . if it is a binary operation is it commutative and associative too?