Question

Consider an algebraic system (Q,*), where Q is the set of all non-zero

rational numbers and * is a binary operation defined by a * b =a + b - ab .

Determine whether (Q,*) is a group.​

Answer 1

$\textbf{Given:}$

* is defined on Q by a*b=a+b-ab

$\text{Take 5 and \frac{5}{4} \in Q}$

$5*\frac{5}{4}=5+\frac{5}{4}-5(\frac{5}{4})$

$5*\frac{5}{4}=\frac{25}{4}-\frac{25}{4}$

$5*\frac{5}{4}=0{\notin}Q$

$\implies\text{Closure axiom is not atisfied}$

$\textbf{Hence (Q,*) is not a group}$

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Show that the set z of all integers form a group with respect to binary operation defined by a*b-a+b+1 is an abelian group

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