Pub Mar 06, 2026 Up Jun 10, 2026
Definition 1. Let $X$ and $Y$ be two sets.
$X$ and $Y$ are equinumerous, written
\[X \equ Y,\]if there exists a bijection from $X$ onto $Y$.
$X$ is not a greater set than $Y$, written
\[X \lequ Y,\]if there exists an injection from $X$ to $Y$.
$X$ is a smaller set than $Y$, written
\[X \lnequ Y,\]if $X \lequ Y$ and $X \not\equ Y$.
Proposition 1 (Reflexivity of $\equ$). For any set $X$,
\[X \equ X.\]Proposition 2 (Symmetry of $\equ$). For any sets $X$ and $Y$,
\[X \equ Y \enspace\rimp\enspace Y \equ X.\]Proposition 3 (Transitivity of $\equ$). For any $X$, $Y$ and $Z$,
\[X \equ Y \,\land\, Y \equ Z \enspace\rimp\enspace X \equ Z.\]To summarize reflexivity, symmetry and transitivity of $\equ$:
Proposition 4. $\equ$ is an equivalence relation.
Proposition 5 (Reflexivity of $\lequ$). For any set $X$,
\[X \lequ X.\]Proposition 6 (Transitivity of $\lequ$). For any sets $X$, $Y$ and $Z$,
\[X \lequ Y \,\land\, Y\lequ Z \enspace\rimp\enspace X\lequ Z.\]Proposition 7 (Schröder-Bernstein Theorem). For any sets $X$ and $Y$,
\[X \lequ Y \,\land\, Y\lequ X \enspace\rimp\enspace X\equ Y.\]Proposition 8. For any sets $X$ and $Y$, if $X \subseteq Y$, then $X \lequ Y$.