Short Answer TypeState whether the following statements are true or false. Justify.
(i) For an arbitrary binary operation * on a set N, a*a = a ∀ a ∈N.
(ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a.
Consider a binary operation * on N defined as a * b = a3 + b3 . Choose the correct answer.
(A) Is * both associative and commutative ?
(B) Is * commutative but not associative?
(C) Is * associative but not commutative?
(D) Is * neither commutative nor associative?
Here. a * b = a3 + b3 = b3 + a3
= b * a for all a, b ∈ N.
∴ (a* b)* c = (a3 + b3) * c= (a3 +b3 )3 + c3
and a * (b * c) = a * (b3 + c3 ) = a3 + (b3 + c3)3 is not associative.
∴ (B) is correct answer.
Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).
Then
(i) Find the identify element of (A, *)
(ii) Find the invertible elements of (A, *)
Define binary operation ‘*’ on Q as follows : a * b = a + b – ab, a, b∈ Q
(i) Find the identity element of (Q, *).
(ii) Which elements in (Q. *) are invertible?
is the inverse of a ≠ 0 for the multiplication operation ‘x’ on R.
Switch
