discrete mathematics - Are these sets equipotent?

I need to decide which of these three sets are equipotent:

$M_1=\{(n_1,n_2,n_3)\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}\ |\ n_1+n_2=n_3\}$

$M_2 = \{M\in P(\mathbb{Z})\ |\ 0\in M\}$

$M_3 = \cup _{a\in\mathbb{Z}}\{x\in\mathbb{R}\ |\ a\leq x < \frac{2a+1}{2}\}$

I want to prove (or disprove) the equipotency by finding injections to and from $\mathbb{N}$, $P(\mathbb{N})$ and $\mathbb{R}$ (Cantor-Schroeder-Bernstein).

I've already proven that $M_1$ is equipotent to $\mathbb{N}$:

1) $M_1\rightarrow\mathbb{N}$, $(n_1,n_2,n_3)\mapsto 2^{n_1}\cdot 3^{n_2}\cdot 5^{n_3}$

2) $\mathbb{N}\rightarrow M_2, n\mapsto (n,n,2n)$

I'm stuck finding injections like this for $M_2$ and $M_3$.

It already seems that $M_2$ is equipotent to $P(\mathbb{N})$ and $M_3$ is equipotent to $\mathbb{R}$, but what are the corresponding injections?

Answer

Since for any set $\;X\in P(\Bbb N)\;$ (for me the naturals do not contain zero) , we have that $\;X\cup\{0\}\in M_2\;$ , so we have that

$$\mathfrak c=|P(\Bbb N)|\le|M_2|\le|P(\Bbb Z)|=\mathfrak c\implies |M_2|=\mathfrak c$$