question_answer 52)

Let * be a binary operation on the set @ of rational numbers as follows :       (i) a * b = a ? b                    (ii) a * b = a2 + b2       (iii) a * b = a + ab                (iv) a * b = (a ? b)2       (v) a * b                      (vi) a * b = ab2         Find which of i.e. binary operations are commutative and which are associative.  

Answer:

(i) Given a * b = a ? b       Since a ? b              is not commutative.       Also (a * b) * c = (a ? b) * c = a ? b ? c       and a * (b * c) = a *(b ? c) = a ? (b ? c)       = a ? b + c       As a ? b ? c a ? b + c              is not associative. (ii)   Given, a * b = a2 = b2,       a * b = a2 + b2 = b2 + a2 = b* a        commutative.       Also (a * b) * c = (a2 + b2) * c       = (a2 + b2)2 + c2       and a * (b * c) = a * (b2 + c2)             = a2 + (b2 + c2)2                              is associative. (iii)  Given a * b = a +       Here, a * b = a + ab       And b * a = b + ba.       But a + ab  b + ba       But a * b  b * a        is not commutative.       Also (a * b) * c = (a + ab) * c             = a + ab + (a + ab) c             = a + ab + ac + abc       and a * (b * c) = a * (b + bc)       = a + a (b + bc)       = a + ab + abc       But a + ab + ac + abc a + ab + abc              is not associative.       (iv)  Given, a * b = (a ? b)2       Here a * b = (a ? b)2 = (b ? a)2       = b * a        is commutative.       Also, (a * b) * c = (a ? b)2 * c             = ((a ? b)2 ? c)2                                  and a * (b * c) = a * (b ? c)2       = (b ? (b ? c)2)2       But ((a ? b)2 ? c)2              is not associative. (b)  Given a * b =       Here,        is commutative.       Also,             and                    is associative. (vi)  Given a * b = ab2 = ab2       Now, a * b = ab2 and b * a = b2a              is not commutative.       Also, (a * b) * c = (ab2) * c = ab2 c2             and  a * (b * c) = a* (bc2)             = a(bc2)2 = ab2 c4                                         Clearly (a * b) * c        is not associative.