Question

Q⁺ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q⁺ is defined as $a\odot =\frac{ab}{2}$, then the inverse of 3 is
(a) $\frac{4}{3}$

(b) 2

(c) $\frac{1}{3}$

(d) $\frac{2}{3}$

Answer 1

(a) $\frac{4}{3}$

Let e be the identity element in Q⁺ with respect to $\odot$ such that

$$\begin{gathered} a*e=a=e*a,\forall a\in Q^{+} \\ a*e=a\text{and}e*a=a,\forall a\in Q^{+} \\ \frac{ae}{2}=a\text{and}\frac{ea}{2}=a,\forall a\in Q^{+} \\ e=2,\forall a\in Q^{+} \end{gathered}$$

Thus, 2 is the identity element in Q⁺ with respect to $\odot$.


$$\begin{gathered} \text{Let}b\in Q^{+}\text{be the inverse of 3}.\text{Then,} \\ \text{3}*b=e=b*3 \\ 3*b=e\text{and}b*3=e \\ \frac{3b}{2}=2\text{and}\frac{b\left(3\right)}{2}=2 \\ b=\frac{4}{3} \\ \text{Thus,}\frac{4}{3}\text{is the inverse of}3. \end{gathered}$$