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

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 – b 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 as

a ∗ b = a – b

We can observed that for 2,3,4 ϵ Q

2 * 3 = 2-3 = -1 and 3 * 2 = 3-2 =1

2 * 3 ≠ 3 * 2

⇒ the operation * is not commutative.

Also, (2 * 3) * 4 = (-1) * 4 = -1 -4 = -5 and

2 * (3 * 4) = 2 * (-1) = 2 – (-1) = 3.

⇒ (2 * 3) * 4 ≠ 2 * (3 * 4)

Therefore, the operation * is not associative.

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

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