question_answer 45)

For each binary operation * defined below, determine whether * is commutative or associative. (i)   On Z, define a * b = a ? b (ii)   On Q, define a * b = ab + 1 (iii)  on Q, define a * b (iv)  On Z+, define a * b = 2ab (v)  On Z+, define a * b = ab (vi)  On R ? {?1} define a * b  

Answer:

(i) Given a * b = a ? b       Here a * b = a ? b and b * a = b ? a       But a ? b  b ?             is not commutative.       Also,                     = a ? b ? c       and              is not associative.       (ii)   Given       Here                          is commutative.       Also (a * b) c = (ab + 1) * c       = (ab + 1) c + 1 = abc + c + 1       and a * (b * c) = a * (bc + 1)             = a(bc + 1) + 1 = abc + a + 1              is not associative.       (iii) Given       Here                    is commutative.       Also,       and              is associative. (iv)  Given       Here a * b = 2ab = 2ba                         = b * a        is commutative.       Also, (a * b) * c (2ab) * c =       and       Clearly              is not associative. (v) Given a * b = ab       Here a * b = ab and b * a = ba              is not commutative.       Also (a * b) * c= (ab) * (ab)c = abc       and a * (b * c) = a * bc =              is not associative. (vi)  Given       Here Clearly a * b is not necessarily equal to b * a. For example,, let a = 2 and b = 3    and       Hence a * b        is not commutative.       Also             and             Clearly, a * (b * c) is not necessarily equal to (a*b)*c       For example.       Let a = 2, b = 3 and c = 4                         and                         Hence * is not associative.