From 500d5382f0160e21fad128f77945a507415279d8 Mon Sep 17 00:00:00 2001
From: Luc <luc@bijl.us>
Date: Wed, 22 Nov 2023 10:19:50 +0100
Subject: [PATCH] Finished logic section.

---
 config/en/mkdocs.yaml        |  7 +++++
 docs/en/mathematics/logic.md | 56 ++++++++++++++++++++++++++----------
 2 files changed, 48 insertions(+), 15 deletions(-)

diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml
index 96f6451..6c9afd5 100755
--- a/config/en/mkdocs.yaml
+++ b/config/en/mkdocs.yaml
@@ -31,6 +31,13 @@ plugins:
 markdown_extensions:
   - pymdownx.arithmatex:
       generic: true
+  - pymdownx.critic
+  - pymdownx.caret
+  - pymdownx.keys
+  - pymdownx.mark
+  - pymdownx.tilde
+  - def_list
+
 
 extra_javascript:
   - javascripts/mathjax.js
diff --git a/docs/en/mathematics/logic.md b/docs/en/mathematics/logic.md
index 808f1ec..8375842 100644
--- a/docs/en/mathematics/logic.md
+++ b/docs/en/mathematics/logic.md
@@ -1,21 +1,47 @@
 # Logic
 
-> *Definition*: a statement is a sentence that is either true or false, never both.
+*Definition*: a statement is a sentence that is either true or false, never both.
 
-> *Definition* **- Logical operators**: let $A$ and $B$ be assertions. 
-> * The assertion $A$ and $B$ ($A \land B$) is true, iff both $A$ and $B$ are true.
-> * The assertion $A$ or $B$ ($A \lor B$) is true, iff at least one of $A$ and $B$ is true.
-> * The negation of $A$ ($\neg A$) is true iff $A$ is false.
+<br>
 
-> *Definition* **- Implies**: if $A$ and $B$ are assertions then the assertion if $A$ then $B$ ($A \implies B$) is true iff
-> * $A$ is true and $B$ is true,
-> * $A$ is false and $B$ is true,
-> * $A$ is false and $B$ is false.
-> 
-> This also works the opposite way, if $B$ then $A$ ($A \Longleftarrow B$)
+*Definition* **- Logical operators**: let $A$ and $B$ be assertions. 
+* The assertion $A$ and $B$ ($A \land B$) is true, iff both $A$ and $B$ are true.
+* The assertion $A$ or $B$ ($A \lor B$) is true, iff at least one of $A$ and $B$ is true.
+* The negation of $A$ ($\neg A$) is true iff $A$ is false.
 
-> *Definition* **- If and only if**: if $A$ and $B$ are assertions then the assertion $A$ if and only if $B$ (A \iff B) is true iff
-> * $(A \Longleftarrow B) \land (a \implies B)$.
->
-> This leads to the following table.
+<br>
 
+*Definition* **- Implies**: if $A$ and $B$ are assertions then the assertion if $A$ then $B$ ($A \implies B$) is true iff
+* $A$ is true and $B$ is true,
+* $A$ is false and $B$ is true,
+* $A$ is false and $B$ is false.
+ 
+This also works the opposite way, if $B$ then $A$ ($A \Longleftarrow B$)
+
+<br>
+
+*Definition* **- If and only if**: if $A$ and $B$ are assertions then the assertion $A$ if and only if $B$ (A \iff B) is true iff
+* $(A \Longleftarrow B) \land (a \implies B)$.
+
+:   This leads to the following table.
+
+    | $A$   | $B$   | $A \implies B$ | $A \Longleftarrow B$ | $A \iff B$|
+    | :---: | :---: | :------------: | :------------------: | :-------: |
+    | true  | true  | true           | true                 | true      | 
+    | true  | false | false          | true                 | false     |
+    | false | true  | true           | false                | false     |
+    | false | false | true           | true                 | true      |
+
+<br>
+
+*Definition*: suppose $P$ and $Q$ are assertions. $P$ implies $Q$ if $P \implies Q$ is true. $P$ and $Q$ are equivalent if $P$ implies $Q$ and $Q$ implies $P$. 
+
+## Methods of proof
+
+*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.
+
+*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 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