From 8f48bec66b0656104958580b0d709dee39d81250 Mon Sep 17 00:00:00 2001 From: eggy Date: Sat, 11 Feb 2023 14:25:46 -0500 Subject: [PATCH] ece108: add cardinality --- docs/1b/ece108.md | 25 +++++++++++++++++++++++++ 1 file changed, 25 insertions(+) diff --git a/docs/1b/ece108.md b/docs/1b/ece108.md index 7992993..e960436 100644 --- a/docs/1b/ece108.md +++ b/docs/1b/ece108.md @@ -597,4 +597,29 @@ If $f: X\to Y$ and $g: Y\to Z$ are bijective: - $gf$ exists and is invertible - $f^{-1}g^{-1}=(gf)^{-1}$ and exists +## Cardinality + +!!! definition + - A **countably infinite** set is such that there exists a function that maps the set to the set of natural numbers. + - A **countable** set is a finite set or a countably infinite set. + - An **uncountable** or **uncountably infinite** set is not countable. + +The **cardinality** of a set is the number of elements in that set. + +$$|S|$$ + +If two sets have a finite number of elements, their Cartesian product will have the same number of elements as the product of their elements. + +$$|A|,|B|\in\mathbb N\implies|A\times B|=|A||B|$$ + +If two sets $X$ and $Y$ have finite cardinality and $f:X\to Y$: + +- An injective function must have $|X|\leq |Y|$. +- A surjective function must have $|X|\geq |Y|$. +- A bijective function occurs if and only if $|X|=|Y|$. + +A set is **finite** if it is empty or it is mappable to a subset of the natural numbers. By definition, the set of natural numbers is infinite. + +$$\exists n\in\mathbb N,\exists f\text{ is bijective}, f:S\to \mathbb N_n,|s|=n$$ +