Question

Discuss the associative property of binary operation *defined on A=Q-[-1] by the rule a*b=a-b+ab for all $a,b\in A$

Answer 1

Given $\ast$ is a binary operation on Q-{1} defined by $a\ast b=a-b+ab$

Associativity :

Let a, b, c $\in$ A

$(a\ast b)\ast c=(a-b+ab)\ast c=(a-b+ab)-c+(a-b+ab)c=a-b+ab-c+ac-bc+abc$

And,

$a\ast (b\ast c)=a\ast (b-c+bc)=a-(b-c+bc)+a(b-c+bc)=a-b+c-bc+ab-ac+abc$

Since, $(a\ast b)\ast c\ne a\ast (b\ast c)$

Hence, $\ast$ is not associative on A.