Let * be a binary operation on N, N being set of natural number define

1.	Let * be a binary operation on N, N being set of natural number defined by a * b = ab for all a, b ∈ N. Verify * is associative and commutative on N.
Answer» Given , binary operation is a * b = a^b Let, 2, 3 ∈ N So, 2 * 3 = 2³ = 8 and 3 * 2 = 3² = 9 2 * 3 ≠ 3 * 2 So, '' is not commutative on N Also, 2 (2 * 3) = 2 * 2³ = 2 * 8 = 2⁸ = 256 and (2 * 2) * 3 = 2² * 3 = 4 * 3 = 4³ = 64 So, 2 * (2 * 3) ≠ (2 * 2) * 3 So, '*' is not associative on N

Let * be a binary operation on N, N being set of natural number defined by a * b = ab for all a, b ∈ N. Verify * is associative and commutative on N.

Answer»

Given , binary operation is a * b = a^b

Let, 2, 3 ∈ N

So, 2 * 3 = 2³ = 8 and 3 * 2 = 3² = 9

2 * 3 ≠ 3 * 2

So, '*' is not commutative on N

Also, 2 * (2 * 3) = 2 * 2³ = 2 * 8 = 2⁸ = 256

and (2 * 2) * 3 = 2² * 3 = 4 * 3 = 4³ = 64

So, 2 * (2 * 3) ≠ (2 * 2) * 3

So, '*' is not associative on N