Prove the following are surjective, or disprove with a counter-example:
- $f\colon \mathbb{Q} \to \mathbb{Q}$, $f(x) = 1 + 2x$.
- $f\colon \mathbb{Z} \to \mathbb{N}\cup\{0\}$, $f(x) = |1 - x|$.
- $f\colon \mathbb{Q} \to \mathbb{Q}$, $f(x) = 4 - 2x^3$.
- $f\colon \mathbb{Z}^2 \to \mathbb{Z}$, $f(x, y) = x + y$.
Please show me the most effective method to lay out such proofs.
These come from a manual on Set Theory, which I am trying to reach to myself. Please be understanding!