Cardinal Number

Definition

$|A|$ The number that represents the size of the number of elements in a set. uses a natural number for a finite set, and uses the infinite cardinal number for an infinite set.

Ordering

$|X|\le |Y|\iff \exists 1-1f:X\to Y$

There exist an injective function $f:X\to Y$

Operations

Let $x,y,z$ be cardinal numbers.

Addition

$\forall X,\forall Y,X\cap Y=\varnothing \Rightarrow |X|+|Y|=|X\cup Y|$

$|X|\cdot |Y|=|X\times Y|$ where $\times$ is a Cartesian Product

$(x+y)+z=x+(y+z)$ Associative Property of addition $x+y=y+x$ Commutative Property of addition

Multiplication

$(x\cdot y)\cdot z=x\cdot (y\cdot z)$ Associative Property of multiplication $x\cdot y=y\cdot x$ Commutative Property of multiplication $x\cdot (y+z)=x\cdot y+x\cdot z$ Distributive Property of multiplication over addition

Cardinal Exponential

$|X{|}^{|Y|}=|X^Y|$ where $X^Y=\{f|f:X\to Y\}$

$|2^X|=2^{|X|}$

$x^y{x}^z=x^{y+z}$ $(x^y{)}^z=x^{yz}$ $(xy{)}^z=x^z{y}^z$

Facts

$\forall A:\exists !|A|$

$\forall a,\exists A,|A|=a$ where $a$ is a cardinal number

$A=\varnothing \iff |A|=0$

$A\approx B\iff |A|=|B|$

$|\mathbb{N}|={\aleph }_0$

$|\mathbb{R}|=\mathfrak{c}$

$(\nexists C\subset A,|C|=|B|)\wedge (\exists D\subset B,|D|=|A|)\Rightarrow |A|<|B|$

Cantor-Schroder-Bernstein Theorem

Definition

$|A|\le |B|\wedge |B|\le |A|\Rightarrow |A|=|B|$

Link to original

$|A|\le |B|\wedge |B|\le |C|\Rightarrow |A|\le |C|$

${\aleph }_0+{\aleph }_0={\aleph }_0$

$\mathfrak{c}+\mathfrak{c}=\mathfrak{c}$

${\aleph }_0+\mathfrak{c}=\mathfrak{c}$

${\aleph }_0{\aleph }_0={\aleph }_0$

$\mathfrak{cc}=\mathfrak{c}$

${\aleph }_0\mathfrak{c}=\mathfrak{c}$

$\mathfrak{c}=2^{{\aleph }_0}={\aleph }_0^{{\aleph }_0}={\mathfrak{c}}^{{\aleph }_0}$

$2^{\mathfrak{c}}={\aleph }_0^{\mathfrak{c}}={\mathfrak{c}}^{\mathfrak{c}}$