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

The Correct Option is D

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.