How is ($\mathbb{Z}\setminus\mathbb{Q}$) a subset of $\mathbb{N}$?

Question

I do not understand why the set ($\mathbb{Z}\setminus\mathbb{Q}$) is a subset of $\mathbb{N}$. $\mathbb{Q}$ extends the $\mathbb{Z}$ by adding fractions. So there cannot be an element in $\mathbb{Z}$ which is not in $\mathbb{Q}$. Where am I going wrong with my thoughts?

Answer 1

Take elements of $\mathbb{Z}$, then remove elements of $\mathbb{Q}$. What's left? Nothing. And for sure, $\varnothing$ is a subset of $\mathbb{N}$. Along the same lines, $\mathbb{Z} \setminus \mathbb{Q}$ is a subset of the set of all pieces of furniture in my house. (For this, you must be okay with the fact that the empty set is a subset of every set.)

Answer 2

Hint: $\mathbb Z\setminus\mathbb Q=\emptyset$ and $\emptyset\subset\mathbb N$.