Q15 of 27 Page 74

For all a, b ∈ N, we define a * b = a3 + b3.

Show that * is commutative but not associative.


let a = 1,b = 2N


a*b = 13 + 23 = 9


And b*a = 23 + 13 = 9


Hence * is commutative.


Let c = 3


(a*b)*c = 9*c = 93 + 33


a*(b*c) = a*(23 + 33) = 1*35 = 13 + 353


(a*b)*c a*(b*c)


Hence * is not associative.


More from this chapter

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13

Show that * on R –{ - 1}, defined by is neither commutative nor associative.

 14

For all a, b ∈ R, we define a * b = |a – b|.

Show that * is commutative but not associative.


 16

Let X be a nonempty set and * be a binary operation on P(X), the power set of X, defined by A * B = A ∩ B for all A, B ∈ P(X).

(i) Find the identity element in P(X).


(ii) Show that X is the only invertible element in P(X).


 17

A binary operation * on the set (0, 1, 2, 3, 4, 5) is defined as


Show that 0 is the identity for this operation and each element a has an inverse (6 - a)