Subset Notation.

Question

I have just read that: $$ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} $$

But surely it is more appropriate that:

$$ \mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} $$

Is true, right?

Answer 1

Let $A,B$ be sets.

We say that $A \subseteq B $ is true, if $a \in A \Rightarrow a \in B$.

Moreover, we say that $A \subset B$, if $A \subseteq B$ and $A \neq B$.

This means that $A \subset B$ is a stronger condition: $A\subset B \Rightarrow A\subseteq B$.

Just like $<$ is a stronger version of $\leq$, e.g.: $1\leq 1,$ but $1 \not\lt 1$

In your case, $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ and $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$ are both true, since there are negative integers, non-integer rationals, and irrationals in $\mathbb Z, \mathbb Q,$ and $ \mathbb R$ respectively.

Answer 2

It is not wrong to say that e.g. $\mathbb{N} \subseteq \mathbb{Z}$, just as it is not wrong to write that e.g. $2 \leq 3$. But it is more precise to write that $\mathbb{N} \subset \mathbb{Z}$, since we know that the inclusion is strict.