Subjects: Set Theory
Links: Natural Numbers, ZF Axioms

Cardinality of Sets

Def: Sets $A$ and $B$ are equipotent (have the same cardinality) if there is a bijective function $f$ with domain $A$ and range $B$. We denote this by $|A|=|B|$.

Th:

• $A$ is equipotent to $A$
• If $A$ is equipotent to $B$, then $B$ is equipotent to $A$
• If $A$ is equipotent to $B$, and $B$ is equipotent to $C$, then $A$ is equipotent to $C$

Note that equipotency is not an equivalence relation, since there cannot exits a set of all sets, which would be required to make the equivalence relation.

Def: The cardinality of $A$ is less than or equal to the cardinality of $B$ (notation: $|A|\le |B|$) if there is an injection from $A$ to $B$.

We also write $|A|<|B|$ if there’s an injective function $f$ from $A$ to $B$, but there’s no bijection. Similarly, the cardinality of $A$ is greater or equal to the cardinality of $B$ (notation: $|A|\ge |B|$) if there’s a surjection form $A$ to $B$

Lemma:

• If $|A|\le |B|$ and $|A|=|C|$, then $|C|\le |B|$
• If $|A|\le |B|$ and $|B|=|C|$, then $|A|\le |C|$
• $|A|\le |A|$
• If $|A|\le |B|$ and $|B|\le |C|$, then $|A|\le |C|$
• If $A\subseteq B$, then $|A|\le |B|$

Lemma: If $A_1\subseteq B\subseteq A$, and $|A|=|A_1|$ , then $|B|=|A|$.

Cantor-Bernstein Theorem
If $|X|\le |Y|$ and $|Y|\le |X|$, then $|Y|=|X|$

An alternative proof of the Cantor-Bernstein Theorem, using monotone functions.

Def: Let $F:\mathcal{P}(A)\to \mathcal{P}(A)$ , a set $X\subseteq A$ is called a fixed point of $F$ if $F(X)=X$. The function is called monotone if $X\subseteq Y\subseteq A$ implies $F(X)\subseteq F(Y)$.

Lemma: Let $F:\mathcal{P}(A)\to \mathcal{P}(A)$ be monotone, then it has a fixed point. Additionally we can construct it by

This is the least fixed point of $F$, meaning if $F(X)=X$, then $\bar{X}\subseteq X$.

This lemma can be used to prove Cantor-Bernstein Theorem

Def: A function $F:\mathcal{P}(A)\to \mathcal{P}(A)$ is continuous if $F[\bigcup _{i\in \mathbb{N}}{X}_i]=\bigcup _{i\in \mathbb{N}}F[X_i]$ holds for any nondecreasing sequence of subsets of $A$. $⟨X_i\mid i\in \mathbb{N}⟩$ is nondrecreasing if $i\le j$, then $X_i\subseteq X_j$.

Th: Let $F:\mathcal{P}(A)\to \mathcal{P}(A)$ be a monotone continuous function, then $\bar{X}=\bigcup _{i\in \mathbb{N}}{X}_i$ is the least fixed point of $F$, where $X_0=\varnothing$, and

Showing us that the proof using the lemma is equivalent to the one using monotone functions.

Finite Sets

Def: A $S$ is called finite if it is equipotent to some natural numbre $n\in \mathbb{N}$. We then define $|S|=n$ and say that $S$ has $n$ elements. A set if infinite if it is not finite.

Lemma: If $n\in \mathbb{N}$, then there’s no bijection from $n$ into a proper subset of $n$

Cor:

• If $n\mathbb{N}em$, then there’s no bijection form $n$ into $m$
• If $|S|=n$ and $|S|=m$, then $n=m$
• $\mathbb{N}$ is infinite

Th: If $X$ is finite and $Y\subseteq X$, then $Y$ is finite. Moreover, $|Y|\le |X|$.

Th: If $X$ is finite and $f$ a function, then $f[X]$ is finite. Moreover, $|f[X]|\le |X|$

Th: If $X$ and $Y$ are finite, then $X\cup Y$ is finite. Moreover, $|X\cup Y|\le |X|+|Y|$, and if $X$ and $Y$ are disjoint, then $|X\cup Y|=|X|+|Y|$.

Th: If $S$ is finite, and if every $X\in S$ is finite, then $\bigcup S$ is finite.

Th: If $X$ is finite, then $\mathcal{P}(X)$ is finite.

Th: If $X$ is infinite, then $|X|>n$ for all $n\in \mathbb{N}$.

Th: If $X$ and $Y$ are finite, then $|X\times Y|$, and $|X^Y|$ are finite, additionally, we get that $|X\times Y|=|X|\cdot |Y|=|Y\times X|$ and $|X^Y|=|X{|}^{|Y|}$.

Th: If $X$ is finite, then $\mathcal{P}(X)$ is finite and $|\mathcal{P}(X)|=2^{|X|}$

Countable Sets

Def: A set $S$ is countable if $|S|=|\mathbb{N}|$. A set $S$ is at most countable if $|S|\le |\mathbb{N}|$

Th: An infinite subset of a countable set is countable.

Cor: A set is at most countable iff it is either finite or countable.

Th: The range of an infinite sequence $⟨a_n{⟩}_{n\in \mathbb{N}}$ is at most countable. In other words, the image of a countable set under any mapping is at most countable.

Th: The union of two countable sets is countable.

Cor: The union of a finite system of countable sets is countable

Th: If $A$ and $B$ are countable, then $A\times B$ is countable.

Cor: The Cartesian product of a finite number of countable sets is countable. Consequently, $\text{Bbb N { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#m">\#m</a>}}$ is countable for every $m>0$.

Cor: Let $⟨A_n\mid n\in \mathbb{N}⟩$ be a countable system of at most countable sets, and let $⟨a_n\mid n\in \mathbb{N}⟩$ be a system of enumerations of $A_n$, i.e., for each $n\in \mathbb{N}$, then $a_n=⟨a_n(k)\mid k\in \mathbb{N}⟩$ is an infinite sequence, and $A_n=\{a_n(k)\mid k\in \mathbb{N}\}$. Then $\bigcup _{n\in \mathbb{N}}{A}_n$ is at most countable.

Th: If $A$ is countable, then the set $\mathrm{Seq}(A)$ of all finite sequences of elements of $A$ is countable

Cor: The set of all finite subsets of a countable set is countable.

Th: An equivalence relation on a countable set has at most countable many equivalence classes

Cor: The set of all integers $\mathbb{Z}$ and the set of all rational numbers $\mathbb{Q}$ are countable.

Th: Let $\mathfrak{U}$ be a structure with the universe $A$, and let $C\subseteq A$ be at most countable. Then $\bar{C}$, the closure of $C$, is also at most countable.

Def: $|A|={\mathrm{\aleph }}_0$, for all $A$ countable

Def: Let $A$ be a set, then we define $[A{]}^n:=\{S\subseteq A\mid |S|=n\}$. We can extend this definition using the infinite ordinal $\omega$, and we can get $[A{]}^{<\omega }:=\bigcup _{n\in \mathbb{N}}[A{]}^n=\{S\subseteq A\mid S\text{ finite}\}$, extending this even further we can have the countably infinite subsets of $A$, denoted as $[A{]}^{\omega }$. Finally if we want all the countable subsets of $A$, it can be denoted as $[A{]}^{\le \omega }$.

Th: Let $A$ be countable, then $[A{]}^n$ is countable for $n>0$