Is the cardinality of an empty set comparable to some other sets by the definition?

Question

I've been learning mathematical analysis written by B.A Zorich recently. The description of Card X ≤ Card Y in this book is "there exists a bijective mapping of X onto some subset of Y". But here comes my question: if X is an empty set, it is well known that the inequality Card X ≤ Card Y holds (Y is an nonempty set here), but is there an bijective mapping of X onto some subset of Y? I mean X is empty, how can we construct the bijective? Incidentally, my English is poor, if I don't make myself clear, please just tell me, I'll try to express it more clearly.

Answer 1

Suppose that $X=\emptyset$. Consider the identity function $\text{id}\colon X\to\emptyset$. It is clear that $\emptyset\subseteq Y$ and that $\text{id}$ is a bijection, because it is vacously injective and surjective.