# 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.

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, $\mathbb{Q}\not\subset\mathbb N$.
$$\frac12\in\mathbb Q\setminus\mathbb N.$$