Determine whether the following operation define a binary operation on

1.	Determine whether the following operation define a binary operation on the given set or not:(i) ‘’ on N defined by a b = ab for all a, b ∈ N.(ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.(iii) ‘’ on N defined by a b = a + b – 2 for all a, b ∈ N
Answer» (i) Given ‘’ on N defined by a b = a^b for all a, b ∈ N. Let a, b ∈ N. Then, a^b∈ N [∵ a^b≠ 0 and a, b is positive integer] ⇒ a * b ∈ N So, a * b ∈ N, ∀ a, b ∈ N Thus, * is a binary operation on N. (ii) Given ‘O’ on Z defined by a O b = a^b for all a, b ∈ Z. Both a = 3 and b = -1 belong to Z. ⇒ a * b = 3^-1 = 1/3 ∉ Z Thus, * is not a binary operation on Z. (iii) Given ‘’ on N defined by a b = a + b – 2 for all a, b ∈ N If a = 1 and b = 1, a * b = a + b – 2 = 1 + 1 – 2 = 0 ∉ N Thus, there exist a = 1 and b = 1 such that a * b ∉ N Therefore, * is not a binary operation on N.

Determine whether the following operation define a binary operation on the given set or not:(i) ‘’ on N defined by a b = ab for all a, b ∈ N.(ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.(iii) ‘’ on N defined by a b = a + b – 2 for all a, b ∈ N

Answer»

(i) Given ‘*’ on N defined by a * b = a^b for all a, b ∈ N.

Let a, b ∈ N. Then,

a^b∈ N [∵ a^b≠ 0 and a, b is positive integer]

⇒ a * b ∈ N

So,

a * b ∈ N, ∀ a, b ∈ N

Thus, * is a binary operation on N.

(ii) Given ‘O’ on Z defined by a O b = a^b for all a, b ∈ Z.

Both a = 3 and b = -1 belong to Z.

⇒ a * b = 3^-1

= 1/3 ∉ Z

Thus, * is not a binary operation on Z.

(iii) Given ‘*’ on N defined by a * b = a + b – 2 for all a, b ∈ N

If a = 1 and b = 1,

a * b = a + b – 2

= 1 + 1 – 2

= 0 ∉ N

Thus, there exist a = 1 and b = 1 such that a * b ∉ N

Therefore, * is not a binary operation on N.