Finished set-theory, updated start menu: added references.

docs/en/mathematics/set-theory/additional-axioms.md

@@ -1 +1,27 @@
 # Additional axioms
+
+## Axiom of choice
+
+> *Principle*: let $C$ be a collection of nonempty sets. Then there exists a map 
+>
+>$$
+>    f: C \to \bigcap_{A \in C} A
+>$$
+>
+> with $f(A) \in A$. 
+>
+> * The image of $f$ is a subset of $\bigcap_{A \in C} A$. 
+> * The function $f$ is called a **choice function**.
+
+The following statements are equivalent to the axiom of choice.
+
+* For any two sets $A$ and $B$ there does exist a surjective map from $A$ to $B$ or from $B$ to $A$.
+* The cardinality of an infinite set $A$ is equal to the cardinality of $A \times A$.
+* Every vector space has a basis.
+* For every surjective map $f: A \to B$ there is a map $g: B \to A$ with $f(g(b)) = b$ for all $b \in B$.
+
+## Axiom of regularity
+
+> *Principle*: let $X$ be a nonempty set of sets. Then $X$ contains an element $Y$ with $X \cap Y = \varnothing$. 
+
+As a result of this axiom any set $S$ cannot contain itself.

docs/en/mathematics/set-theory/cardinalities.md

@@ -1 +1,67 @@
 # Cardinalities
+
+## Cardinality
+
+> *Definition*: two sets $A$ and $B$ have the same **cardinality** if there exists a bijection from $A$ to $B$. 
+
+For example, two finite sets have the same cardinality if and only if they have the same number of elements. The sets $\mathbb{N}$ and $\mathbb{Z}$ have the same cardinality, consider the map $f: \mathbb{N} \to \mathbb{Z}$ defined by $f(2n) = n$ and $f(2n+1) = -n$ with $n \in \mathbb{N}$, which may be observed to be a bijection.
+
+> *Theorem*: having the same cardinality is an equivalence relation.
+
+??? note "*Proof*:"
+
+    Let $A$ be a set. Then the identity map is a bijection from $A$ to itself, so $A$ has the same cardinality as $A$. Therefore we obtain reflexivity.
+
+    Suppose $A$ has the same cardinality as $B$. Then there is a bijection $f: A \to B$. Now $f$ has an inverse $f^{-1}$, which is a bijection from $B$ to $A$. So $B$ has the same cardinality as $A$, obtaining symmetry.
+
+    Suppose $A$ has the same cardinality as $B$ and $B$ the same cardinality as $C$. So, there exist bijections $f: A \to B$ and $g: B \to C$. Then $g \circ f: A \to C$ is a bijection from $A$ to $C$. So $A$ has the same cardinality as $C$, obtaining transitivity. 
+
+## Countable sets
+
+> *Definition*: a set is called **finite** if it is empty or has the same cardinality as the set $\mathbb{N}_n := \{1, 2, \dots, n\}$ and **infinite** otherwise. 
+
+<br>
+
+> *Definition*: a set is called **countable** if it is finite or has the same cardinality as the set $\mathbb{N}$. An infinite set that is not countable is called **uncountable**.
+
+<br>
+
+> *Theorem*: every infinite set contains an infinite countable subset.
+
+??? note "*Proof*:"
+
+    Suppose $A$ is an infinite set. Since $A$ is infinite, we can start enumerating the elements $a_1, a_2, \dots$ such that all the elements are distinct. This yields a sequence of elements in $A$. The set of all elements in this sequence form a countable subset of $A$.
+
+> *Theorem*: let $A$ be a set. If there is a surjective map from $\mathbb{N}$ to $A$ then $A$ is countable.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Uncountable sets
+
+> *Lemma*: the set $\{0,1\}^\mathbb{N}$ is uncountable.
+
+??? note "*Proof*:"
+
+    let $F: \mathbb{N} \to \{0,1\}^\mathbb{N}$. By $f_i$ we denote the function $F(i)$ from $\mathbb{N}$ to $\{0,1\}$. ...
+
+The power set of $\mathbb{N}$ has the same cardinality as $\{0,1\}^\mathbb{N}$ therefore it also uncountable.
+
+> *Lemma*: the interval $[0,1)$ is uncountable.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Theorem*: $\mathbb{R}$ is uncountable.
+
+??? note "*Proof*:"
+
+    as $\mathbb{R}$ contains the uncountable subset $[0,1)$, it is uncountable.
+
+## Cantor-Schröder-Bernstein theorem
+
+> *Theorem*: let $A$ and $B$ be sets and assume that there are two maps $f: A \to B$ and $g: B \to A$ which are injective. Then there exists a bijection $h: A \to B$. 
+> 
+> Therefore $A$ and $B$ have the same cardinality.

docs/en/mathematics/set-theory/recursion-induction.md

@@ -61,3 +61,39 @@ Hence if the claim holds for some $k \in \mathbb{N}$ then it also holds for $k+1
 $$
     \sum_{i=1}^n i = \frac{n}{2}(n+1).
 $$
+
+> *Principle* **- Strong induction**: suppose $P(n)$ is a predicate for $n \in \mathbb{Z}$, let $b \in \mathbb{Z}$. If the following holds
+>
+> * $P(b)$ is true,
+> * for all $k \in \mathbb{Z}$ we have that $P(b), P(b+1), \dots, P(k-1)$ and $P(k)$ together imply $P(k+1)$. 
+> 
+> Then $P(n)$ is true for all $n \geq b$. 
+
+For example, we claim for the recursion
+
+$$
+\begin{align*}
+    &a_1 = 1, \\
+    &a_2 = 3, \\
+    &a_n = a_{n-2} + 2 a_{n-1}
+\end{align*}
+$$
+
+that $a_n$ is odd $\forall n \in \mathbb{N}$. 
+
+We first check the claim for for $n=1$ and $n=2$, from the definition of the recursion it may be observed that the it is true.
+
+Now suppose that for some $i \in \{1, \dots, k\}$ $a_i$ is odd.
+
+Then by assumption
+
+$$
+\begin{align*}
+    a_{k+1} &= a_{k-1} + 2 a_k, \\
+            &= a_{k-1} + 2 a_{k} + 2(a_{k-2} + 2a_{k-1}), \\
+            &= 2 (a_k + a_{k-2} + 2 a_{k-1}) + a_{k-1},
+\end{align*}
+$$
+
+so $a_{k+1}$ is odd.
+

docs/en/mathematics/start.md

@@ -2,6 +2,7 @@
 
 Welcome to the mathematics page. Some special mathematical environments that will be used in this section are listed and explained below.
 
+* *Principles*: not yet defined.
 * *Definitions* : a precise and unambiguous description of the meaning of a mathematical term. It char-
 acterizes the meaning of a word by giving all the properties and only those properties that must be
 true.
@@ -14,5 +15,11 @@ to proving a theorem.
 that this is a corollary to Theorem A).
 * *Proofs* : a convincing argument that a certain mathematical statement is necessarily true. A proof
 generally uses deductive reasoning and logic but also contains some amount of ordinary language.
-* *Examples* : examples help to understand the meaning of a definition, or the impact of a result.
 * *Algorithms* : recipes to do calculations.
+
+The mathematics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
+
+* The definitions of the special mathematical environments on this page and the sections of logic, set-theory and number-theory are based on the lectures and lecture notes of Hans Cuypers. 
+* The section of calculus is based on the lectures of Luc Habets and the book Calculus by Robert Adams.
+* The section of linear algebra is based on the lectures of Rik Kaasschieter and the book Linear Algebra by Steven Leon.
+* The sections of multivariable calculus and ordinary differential equations are based on the lectures and lecture notes of Georg Prokert and the book Calculus by Robert Adams.