Updated math section.

config/en/mkdocs.yaml

@@ -41,6 +41,7 @@ nav:
   - 'Welcome': index.md
   - 'Mathematics':
     - 'Start': mathematics/start.md
+    - 'Logic': mathematics/logic.md
     - 'Calculus': 
       - 'Limits': mathematics/calculus/limits.md
       - 'Continuity': mathematics/calculus/continuity.md
@@ -68,7 +69,6 @@ nav:
       - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md
       - 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md
 
-
   - 'Physics':
     - 'Start': physics/start.md
     

docs/en/mathematics/logic.md

@@ -0,0 +1,21 @@
+# Logic
+
+> *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.
+
+> *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* **- 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.
+

docs/en/mathematics/start.md

@@ -1,3 +1,18 @@
 # Mathematics
 
-Welcome to the mathematics page.
+Welcome to the mathematics page. Some special mathematical environments that will be used in this section are listed and explained below.
+
+* *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.
+* *Theorems* : a mathematical statement that is proved to be true using rigorous mathematical reasoning. In
+a mathematical text, the term theorem is often reserved for the most important results.
+* *Propositions* : an often interesting result, but generally less important than a theorem.
+* *Lemmas* : a minor result whose purpose is to help in proving a theorem. It is a stepping stone on the path
+to proving a theorem.
+* *Corollaries* : a result in which the (usually short) proof relies heavily on a given theorem (we often say
+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.