From 8a643c5b1b0054dcfb6160e8eaae0f9be9a9562f Mon Sep 17 00:00:00 2001
From: Luc <luc@bijl.us>
Date: Sat, 2 Dec 2023 13:39:20 +0100
Subject: [PATCH] Updated logic and added set theory.

---
 config/en/mkdocs.yaml                  |  2 ++
 docs/en/mathematics/logic.md           | 14 +++++++++----
 docs/en/mathematics/set-theory/sets.md | 27 ++++++++++++++++++++++++++
 3 files changed, 39 insertions(+), 4 deletions(-)
 create mode 100644 docs/en/mathematics/set-theory/sets.md

diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml
index 6c9afd5..d6ed6c9 100755
--- a/config/en/mkdocs.yaml
+++ b/config/en/mkdocs.yaml
@@ -49,6 +49,8 @@ nav:
   - 'Mathematics':
     - 'Start': mathematics/start.md
     - 'Logic': mathematics/logic.md
+    - 'Set theory':
+      - 'Sets': mathematics/set-theory/sets.md
     - 'Calculus': 
       - 'Limits': mathematics/calculus/limits.md
       - 'Continuity': mathematics/calculus/continuity.md
diff --git a/docs/en/mathematics/logic.md b/docs/en/mathematics/logic.md
index cd4530a..390d7a3 100644
--- a/docs/en/mathematics/logic.md
+++ b/docs/en/mathematics/logic.md
@@ -41,10 +41,16 @@
 
 ## Methods of proof
 
-*Direct proof*: for proving $P \implies Q$ only consider the case where $P$ is true.
+> *Direct proof*: for proving $P \implies Q$ only consider the case where $P$ is true.
 
-*Proof by contraposition*: proving $P \implies Q$ to be true by showing that $\neg Q \implies \neg P$ is true.
+<br>
 
-*Proof by contradiction*: using the equivalence of $P \implies Q$ and $\neg Q \implies \neg P$ by assuming $P$ is not true and deducing a contradiction with some obviously true statement $Q$.
+> *Proof by contraposition*: proving $P \implies Q$ to be true by showing that $\neg Q \implies \neg P$ is true.
 
-*Proof by cases*: dividing a proof into cases which makes use of the equivalence of $(P \lor Q) \implies R$ and $(P \implies R) \land (Q \implies R)$. Which together cover all situations under consideration.
\ No newline at end of file
+<br>
+
+> *Proof by contradiction*: using the equivalence of $P \implies Q$ and $\neg Q \implies \neg P$ by assuming $P$ is not true and deducing a contradiction with some obviously true statement $Q$.
+
+<br>
+
+> *Proof by cases*: dividing a proof into cases which makes use of the equivalence of $(P \lor Q) \implies R$ and $(P \implies R) \land (Q \implies R)$. Which together cover all situations under consideration.
\ No newline at end of file
diff --git a/docs/en/mathematics/set-theory/sets.md b/docs/en/mathematics/set-theory/sets.md
new file mode 100644
index 0000000..559a457
--- /dev/null
+++ b/docs/en/mathematics/set-theory/sets.md
@@ -0,0 +1,27 @@
+# Sets
+
+## Sets and subsets
+
+> *Definition*: a set is a collection of elements uniquely defined by these elements. 
+
+Examples are $\mathbb{N}$, the set of natural numbers. $\mathbb{Z}$, the set of integers. $\mathbb{Q}$, the set of rational numbers. $\mathbb{R}$, the set of real numbers and $\mathbb{C}$ the set of complex numbers.
+
+<br>
+
+> *Definition*: suppose $A$ and $B$ are sets. Then $A$ is called a subset of $B$, if for every element $a \in A$ there also is $a \in B$. Then $B$ contains $A$ and can be denoted by $A \subseteq B$. 
+
+The extra line under the symbol implies properness. A subset $A$ of a set $B$ which is not the empty set $\empty$ nor the full set $B$ is called a proper subset of $B$, denoted by $A \subsetneq B$. For example $\mathbb{N} \subsetneq \mathbb{Z}$.
+
+<br>
+
+> *Definition*: if $B$ is a set, then $\wp(B)$ denotes the set of all subsets $A$ of $B$. The set $\wp(B)$ is called the power set of $B$. 
+
+Suppose for example that $B = {x,y,z}$, then $\wp(B) = \{\empty,\{x\},\{y\},\{z\},\{x,y\},\{x,z\},\{y,z\},\{x,y,z\}\}$.
+
+<br>
+
+> *Proposition*: let $B$ be a set with $n$ elements. Then its power set $\wp(B)$ contains $w^n$ elements.
+
+??? note "*Proof*:"
+
+    Let $B$ be set with $n$ elements. A subset $A$ of $B$ is completely determined by its elements. For each element $b \in B$ there are two options, it is in $A$ or it is not. So, there are $2^n$ options and thus $2^n$ different subsets $A$ of $B$.
\ No newline at end of file