Question

On Z, define * by (m * n) = mⁿ + n^m : ∀m, n ∈ Z Is * binary on Z?

Answer 1

No.

* is not a binary operation on Z.

Reason: Since m, n ∈ Z.

So, m, n can be negative also.

Now, if n is negative (Le.) say n = – k where k is +ve.

Then mⁿ = m^–k = `1/"m"^"k"` ∈ Z.

Similarly, when m is negative then n^m ∉ Z.

∴ m * n ∉ Z.

⇒ * is not a binary operation on Z.