real analysis - Problem about bijective function

which of the following statement is True ?

$1.$ There exist a bijective function from $\mathbb{R} - \mathbb{Q}$ to $\mathbb{R}$

$2$ There exist a bijective function from $\mathbb{Q} $ to $\mathbb{Z} \times \mathbb{N}$

$3$ There does exists strictly increasing function from $\mathbb{Z}$ to $\mathbb{N}$

$4$.There does exists strictly increasing onto function from $\mathbb{Z}$ to $\mathbb{N}$

I thinks only option $2 $ is true because countable map to countable

Answer

1) is true since $\mathbb R \setminus \mathbb Q$ has same cardinality as $\mathbb R$

2) is true.

3) and 4) are false. Suppose there is a strictly incerasing function $f: \mathbb Z \to \mathbb N$. Let $f(0)=n$. Then $f(-1), f(-2),....$ satisfy the inequalities $... f(-n). But there are only a finite number of integers less than $n$ in $\mathbb N$. Hence such an $f$ cannot exist.