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Given:
\( (a,b)\circ(c,d) = (ac, bd), \quad (a,b),(c,d)\in \mathbb{R}_0 \times \mathbb{R}_0 \)
i. Commutativity:
\( (a,b)\circ(c,d) = (ac, bd) = (ca, db) = (c,d)\circ(a,b) \)
✔ Commutative
Associativity:
LHS:
\( [(a,b)\circ(c,d)]\circ(e,f) = (ac,bd)\circ(e,f) = (ace, bdf) \)
RHS:
\( (a,b)\circ[(c,d)\circ(e,f)] = (a,b)\circ(ce,df) = (ace, bdf) \)
✔ Associative
ii. Identity Element:
Let identity be \( (x,y) \)
\( (a,b)\circ(x,y) = (a,b) \Rightarrow (ax, by) = (a,b) \)
Thus:
\( x=1,\; y=1 \)
✔ Identity = (1,1)
iii. Inverse Element:
Let inverse of \( (a,b) \) be \( (c,d) \)
\( (a,b)\circ(c,d) = (1,1) \Rightarrow (ac, bd) = (1,1) \)
\( c=\frac{1}{a},\; d=\frac{1}{b} \)
✔ Inverse of \( (a,b) = \left(\frac{1}{a}, \frac{1}{b}\right) \)
✔ Every element is invertible
Conclusion:
✔ Commutative & Associative
✔ Identity = (1,1)
✔ Inverse of (a,b) = (1/a, 1/b)