State whether the following statements are true or false. Justify. (i

1.	State 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 N, then a * (b * c) = (c * b) * a(iii) For every binary operation defined on a set having exactly one element a is necessarily commutative and associative .with a as the identity element and a being the inverse of a.
Answer» (i) If a * b = a + b, then a * a = a + a = 2a ≠ a ∴ statement is false (ii) Since commutative a * (b * c) = (b * c) * a = (c * b) * a hence true. (iii) A = {a} * : A x A → A defined by a * a = a ∀ a ∈ A Also from the table it is commutative and associative. Also a is the identity element and also inverse of a.

State 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 N, then a * (b * c) = (c * b) * a(iii) For every binary operation defined on a set having exactly one element a is necessarily commutative and associative .with a as the identity element and a being the inverse of a.

Answer»

(i) If a * b = a + b, then

a * a = a + a = 2a ≠ a

∴ statement is false

(ii) Since commutative

a * (b * c) = (b * c) * a

= (c * b) * a hence true.

(iii) A = {a} * : A x A → A

defined by a * a = a ∀ a ∈ A

Also from the table it is commutative and associative.

Also a is the identity element and also inverse of a.