‘*’ is a binary operation on R defined as a * b = 2ab1. Determine

1.	‘’ is a binary operation on R defined as a b = 2ab1. Determine whether * is commutative and associative2. Find the identity element if exists.3. Find the inverse element, if exists
Answer» 1. a * b = 2ab = 2ba = b * a Therefore commutative a * (b * c) = a * (2bc) = 4abc (a * b) * c = (2ab) * c = 4 abc Therefore is associative. 2. a * e = a ⇒ 2ae = a ⇒ e = \(\frac{1}{2}\) e * a = a ⇒ 2ea = a ⇒ e = \(\frac{1}{2}\) Therefore identity element is \(\frac{1}{2}\). 3. a * b = \(\frac{1}{2}\) ⇒ 2ab = \(\frac{1}{2}\) ⇒ b = \(\frac{1}{4}\) a, a ≠ 0.

‘’ is a binary operation on R defined as a b = 2ab1. Determine whether * is commutative and associative2. Find the identity element if exists.3. Find the inverse element, if exists

Answer»

1. a * b = 2ab = 2ba = b * a

Therefore commutative

a * (b * c) = a * (2bc) = 4abc

(a * b) * c = (2ab) * c = 4 abc

Therefore is associative.

2. a * e = a ⇒ 2ae = a ⇒ e = \(\frac{1}{2}\)

e * a = a ⇒ 2ea = a ⇒ e = \(\frac{1}{2}\)

Therefore identity element is \(\frac{1}{2}\).

3. a * b = \(\frac{1}{2}\) ⇒ 2ab = \(\frac{1}{2}\) ⇒ b = \(\frac{1}{4}\) a, a ≠ 0.