# Set of natural and rational numbers

Just a quick question: Is it correct to say that the set of rational numbers cannot be a subset of the set of natural numbers? Certainly, we know these two sets have the same cardinality and there exist a bijection between them. But yet, we cannot say the set of rational numbers is contained in (or is a subset of) the set of natural numbers right? Since, after all, we certainly can find a rational number that does not belong in the set of natural numbers. I think I just am mixing up some simple notions of subset or equal sets vs isomorphic sets under some bijective mapping but it's confusing me.

$$\frac12\in\mathbb Q\setminus\mathbb N.$$
It seems to me that your confusion is between $A\subset B$ and $\lvert A\rvert\leqslant\lvert B\rvert$ (that is, the cardinal of $A$ is smaller than or equal to the cardinal of $B$). In turns out that $A\subset B\implies\lvert A\rvert\leqslant\lvert B\rvert$, but this is not an equivalence.
In your specific situation, $\mathbb Q$ and $\mathbb N$ have the same cardinal, but, as you know, $\mathbb{Q}\not\subset\mathbb N$.