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Check the commutativity and associativity of each of the following binary operations: (ix) ‘*’ on Q defined by a * b = (a – b)2 for all a, b ∈ Q (x) ‘*’ on Q defined by a * b = a b + 1 for all a, b ∈ Q

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(ix)  to check: commutativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }{{\left( a\text{ }-\text{ }b \right)}^{2}} \\ =\text{ }{{\left( b\text{ }-\text{ }a \right)}^{2}} \\ =\text{ }b\text{ }*\text{ }a \\ a\text{ }*\text{ }b\text{ }=\text{ }b\text{ }*\text{ }a,\text{ }for\text{ }all\text{ }a,\text{ }b\text{ }\in \text{ }Q \\ \end{array}\]

Thus, * is commutative on Q

to prove: associativity of * on Q

Let a, b, c ∈ Q, then

   

Thus, * is not associative on Q.

(x) to check : commutativity of *

   

Thus, * is commutative on Q

Now we have to prove associativity of * on Q

   

Thus, * is not associative on Q.