Let ∗ be a binary operation on the set Q of rational numbers as follows:
a ∗ b = a + ab

Find which of the binary operations are commutative and which are associative.

Question

Let ∗ be a binary operation on the set Q of rational numbers as follows: a ∗ b = a + ab Find which of the binary operations are commutative and which are associative.

Philoid · Accepted Answer

It is given that ∗ be a binary operation on the set Q of rational numbers is defined as

a ∗ b = a + ab

We can see that 1 * 2 = 1 + 1 × 2 = 1 + 2 = 3

and 2 * 1 = 2 + 2 × 1 = 2+2 = 4

⇒ 1 *2 ≠ 2 * 1: where 1,2 ϵ Q.

⇒ the operation * is not commutative.

Also, We can see that (1 * 2) * 3 ≠ 1 *(2 * 3), where 1,2,3 ϵ Q

Therefore, the operation * is not associative.

Let ∗ be a binary operation on the set Q of rational numbers as follows:a ∗ b = a + abFind which of the binary operations are commutative and which are associative.

Let ∗ be a binary operation on the set Q of rational numbers as follows:
a ∗ b = a + ab

Find which of the binary operations are commutative and which are associative.