Question

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

For an arbitrary binary operation * on a set N, a * a = ∀ a  a * N.

Answer 1

Define an operation * on N as:

a * b = a + b ∀ a, b ∈ N

Then, in particular, for b = a = 3, we have:

3 * 3 = 3 + 3 = 6 ≠ 3

Therefore, statement (i) is false.