9. 1 ⟹ ⟹ x = c log y ½ ½ 10. ½ 1 ½ Getting 11. 1 Equating the arguments and getting x = 1 12. ½ ½ 1 Let x1, x2such that f(x1) = f(, x2)x1,= x2 13. f is one one. 1 ½ Let y such that y= x such that f is onto. (OR) for proving * is commutative 1 ½ for proving * is associative Let (e1, e2) be the identity element. 1 For any element (a,b) , (a,b) * (e1, e2) =(a,b) ) = (a,b) (e1, e2) = (0,0) e1= 0 , e2 = 0 1 ½ But (0,0) There is no identity element for * 1 ½
9. ⟹ ⟹ x = c log y 1 ½ ½ 10. ½ 1 ½ 11. Getting Equating the arguments and getting x = 1 1 12. ½ ½ 1 13. Let x1, x2such that f(x1) = f(, x2)x1,= x2 f is one one. Let y such that y= x such that f is onto. (OR) for proving * is commutative for proving * is associative Let (e1, e2) be the identity element. For any element (a,b) , (a,b) * (e1, e2) =(a,b) ) = (a,b) (e1, e2) = (0,0) e1= 0 , e2 = 0 But (0,0) There is no identity element for * 1 ½ 1 ½ 1 1 ½ 1 ½
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