Question

Check whether the function \( f : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} \) defined as \( f(x, y) = (2y, 3x) \) is injective (one-to-one) or not.

Hint:

Even though this function is injective, it is not surjective over integers! For instance, look at the output pair \((1, 1)\). To get \(2y = 1\), \(y\) would have to be \(0.5\), which is not an integer. So it's an injection but not a bijection!

Answer 1