Equinumerosity

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 $Z$,

\[X \equ Y \,\land\, Y \equ Z \enspace\rimp\enspace X \equ Z.\]

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 $X \lequ Y$.