Definition:Smaller Set/Definition 1

Definition

Let $S$ and $T$ be sets.

$S$ is defined as being smaller than $T$ if and only if there exists a bijection from $S$ to a subset of $T$.

Notation

$S$ is smaller than $T$ can be denoted:

$S \le T$

Some sources denote this using an explicit ordering on the cardinalities of the sets in question:

$\card S \le \card T$

Also see

• Equivalence of Definitions of Smaller Set

• Results about smaller set can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 4$ Larger and smaller