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Given:
\( (a,b)*(c,d) = (a+c, b+d), \quad (a,b),(c,d)\in \mathbb{R}\times\mathbb{R} \)
Commutativity:
\( (a,b)*(c,d) = (a+c, b+d) = (c+a, d+b) = (c,d)*(a,b) \)
✔ Commutative
Associativity:
LHS:
\( [(a,b)*(c,d)]*(e,f) = (a+c, b+d)*(e,f) = (a+c+e, b+d+f) \)
RHS:
\( (a,b)*[(c,d)*(e,f)] = (a,b)*(c+e, d+f) = (a+c+e, b+d+f) \)
✔ Associative
Identity Element:
Let identity be \( (x,y) \)
\( (a,b)*(x,y) = (a,b) \Rightarrow (a+x, b+y) = (a,b) \)
Thus:
\( x=0,\; y=0 \)
✔ Identity = (0,0)
Conclusion:
✔ Commutative & Associative
✔ Identity element = (0,0)