# 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, $\mathbb{Q}\not\subset\mathbb N$.

• By "this is not an equivalence", you mean the converse of that implication is not true in general yea. Think it is clearer to me now thanks! Sep 12, 2018 at 18:07
• @user82479 Yes, that is what I mean. Sep 12, 2018 at 18:08

$$\frac12\in\mathbb Q\setminus\mathbb N.$$

• Certainly, this is easy to see but I am just having difficulty resolving how we can say both sets are isomorphic and yet the rationals are not a subset of the naturals? So the first involves mapping whereas the second involves just elements, thus it doesn't make sense to compare these two questions? Sep 12, 2018 at 18:01
• @user82479: a bijection does not require the the two sets be "compatible", i.e. to share elements.
– user65203
Sep 12, 2018 at 18:03