Question

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

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

Answer 1

Answer: (D)

a + b + ab

Answer 2

Let us assume $a \ast b = a + b + ab$
Now, $a \ast ( b \ast c ) = a \ast ( b + c + bc)$
$= a + b + c + ab + ac + abc$
$= a + b + ab + c + ac + abc$
$= (a \ast b)\ast c$
$\therefore$ It is associative
Now $a \ast b = a+ b + ab$
$= b + a + ba$
$= b \ast a$
Also, it is commutative.
$\therefore$ Our assuming operation is true.