Question

On the set $Q$of all rational numbers the operation $*$ which is both associative and commutative is given by $a*b$, is

• A. $a+b+ab$
• B. $a^2+b^2$
• C. $ab+1$
• D. $2a+3b$

Answer 1

The correct option is A

$a+b+ab$

Explanation for the correct option:

Checking the associative and commutative properties for the expression:

Let us assume, from the first option

$\mathrm{a}*\mathrm{b}=\mathrm{a}+\mathrm{b}+\mathrm{ab}$

Now, from the above expression

$$\begin{gathered} \mathrm{a}*\left(\mathrm{b}*\mathrm{c}\right)=\mathrm{a}*\left(\mathrm{b}+\mathrm{c}+\mathrm{bc}\right) \\ =\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{ab}+\mathrm{ac}+\mathrm{abc} \\ =\mathrm{a}+\mathrm{b}+\mathrm{ab}+\mathrm{c}+\mathrm{ac}+\mathrm{abc} \\ =\left(\mathrm{a}*\mathrm{b}\right)*\mathrm{c} \end{gathered}$$

So, it is associative,

And now,

$$\begin{gathered} \mathrm{a}*\mathrm{b}=\mathrm{a}+\mathrm{b}+\mathrm{ab} \\ =\mathrm{b}+\mathrm{a}+\mathrm{ba} \\ =\mathrm{b}*\mathrm{a} \end{gathered}$$

Also, it is commutative.

Therefore, the correct answer is option (A).