Would all natural numbers be a perfect subset of quotients.?

Question

Would it be more appropriate to write..

$$\mathbb{N} \subseteq \mathbb{Q} $$ As oppose to... $$\mathbb{N} \subset \mathbb{Q}? $$

Because all natural numbers can be expressed a quotient, $\frac{p}{1} $, where $p, \in \mathbb{N} $, right?

Applying this to a specific question,

If $ 3x \in \mathbb{Q} $, then does this only mean that $x \in \mathbb{Q} $, or can $x \in \mathbb{N}? $

Answer 1

The natural numbers form a proper subset of the rational numbers, i.e. it is a subset and different. This is sometimes stressed by using $\subset$ instead of $\subseteq$, but both are correct.

$3x$ for natural $x$ is a natural number, which is a rational number as well.

Answer 2

I think you are having some confusion of the notion of subsets.

Every natural number is as a rational number. So $\mathbb N$ is a subset of $\mathbb Q$. But not every rational number (for example $\frac 23$) is a natural number so $\mathbb N \ne \mathbb Q$. So $\mathbb N$ is a proper subset of $\mathbb Q$.

The statement $A \subseteq B$ means that $A$ is a subset of $B$ and it is possible that $A = B$.

The statement $A \subsetneq B$ means that $A$ is a proper subset of $B$ and $A \ne B$.

The statement $A \subset B$ is acceptably ambiguous. It means $A$ is a subset of $B$. Some people interpret it to mean it is a proper subset and equality is not acceptable [In other words they take "$\subset$" and "$\subsetneq$" to mean the same thing]. While others (myself included) just take it to mean that it is a subset and equality is possible [In other words we take "$\subset$" and "$\subseteq$" to mean the same thing].

Anyway all three of the following statements are true:

$\mathbb N \subset \mathbb Q$

$\mathbb N \subseteq \mathbb Q$

$\mathbb N \subsetneq \mathbb Q$

and $\mathbb N \subsetneq \mathbb Q$ is the strongest least ambiguous statement.

In my opinion, $\mathbb N \subset \mathbb Q$ is the most appropriate to write as, it is a proper subset, but the inequality doesn't need to be emphasized. If one wishes to draw specific attention to the fact that $A \ne B$ then one should use $A \subsetneq B$ but if one doesn't need to draw attention (or if one doesn't know) then either $A \subset B$ or $A \subseteq B$ is fine.

You final question $a \in A$ and $B \subset A$ means that $a \in A$. It is possible that $a \in B$ but we don't know if it is or isn't. So $3x \in \mathbb Q$ means that $3x$ is rational. There is no reason to assume if isn't a natural number as well.

If we wanted to say $3x$ is rational but is not a natural number we couls state: "$3x \in \mathbb Q$ and $3x \not \in \mathbb N$" or "$3x \in \mathbb Q \setminus \mathbb N$". But if you don't specify that $3x$ can not be a natural number, we should assume it might be.