Question

Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.

Answer 1

Let a, b ∈ R. Then,

a + b ∈ R ......(Addition is a binary operation on R)

⇒ a + (a+ b) ∈ R .........(Addition is a binary operation on R)

⇒ 2a + b ∈ R 

Thus, a*b ∈ R  for all a, b ∈ R.

Hence, * is a binary operation on R.

Let a, b,c ∈ R

(a * b) * c = (2a + b) * c = 2 (2a+ b) + c = 4a + 2b + c

a* (b * c) = a* (2b + c) = 2a + 2b + c

Since (a * b) * c ≠ a * (b * c),

Therefore, binary operation * is not associative.