ece108: add cardinality

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.
+
+$$\exists n\in\mathbb N,\exists f\text{ is bijective}, f:S\to \mathbb N_n,|s|=n$$
+