Order of cardinals

Definition

Let $A$ and $B$ be sets. The cardinality of $A$ is at most the cardinality of $B$, written $|A| \le |B|$, when there is an injective function $f \colon A \to B$. Write $|A| < |B|$ when $|A| \le |B|$ and $|A| \neq |B|$.

Remarks

The relation does not depend on the choice of $A$ and $B$ in their classes of equinumerous sets. It is reflexive and transitive. It is antisymmetric by the Schröder–Bernstein theorem. With the axiom of choice, any two cardinals are comparable. See comparability of cardinals.

Explanations

L1 human

$|A| \le |B|$ means that $A$ fits into $B$ without two elements landing in the same place. For example, $|\mathbb{N}| \le |\mathbb{R}|$, because $\mathbb{N}$ is a subset of $\mathbb{R}$.