Cardinality

Equinumerosity

We say set $A$ is equinumerous to set $B$, written ($A\approx B$) if and only if there exists a one-to-one function from $A$ onto $B$.

Power Sets

No set is equinumerous to its power set. This is typically shown using a diagonalization argument.

Equivalence Concept

For any sets $A$, $B$, and $C$:

• $A\approx A$;
• if $A\approx B$, then $B\approx A$;
• if $A\approx B$ and $B\approx C$, then $A\approx C$.

Notice though that $\{⟨A,B⟩\mid A\approx B\}$ is not an equivalence relation since the equivalence concept of equinumerosity concerns all sets.

Finiteness

A set is finite if and only if it is equinumerous to a natural number. Otherwise it is infinite.

Pigeonhole Principle

No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.

Likewise, any set equinumerous to a proper subset of itself must be infinite.

Cardinal Numbers

A cardinal number is a set that is $\mathrm{card}A$ for some set $A$. The set $\mathrm{card}A$ is defined such that

• For any sets $A$ and $B$, $\mathrm{card}A=\mathrm{card}B$ iff $A\approx B$.
• For a finite set $A$, $\mathrm{card}A$ is the natural number $n$ for which $A\approx n$.

Bibliography

• Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).