ece108: add reflexivity, symmetry

docs/1b/ece108.md

@@ -657,6 +657,16 @@ For $R\subseteq X\times Y$:
 
 Relations are trivially proven to be relations through subset analysis.
 
+!!! example
+    For the relation $L$\subseteq R^2=\{\left<x,y\right>\in\mathbb R^2|x<y\}$:
+    
+    Clearly it is a subset of $R^2$, so it is a relation.
+    
+    - The domain is $\mathbb R$.
+    - The range is $\mathbb R$.
+    - $L(\{1,4\})=\{y>4|y\in\mathbb R\}$ (1 OR 4)
+    - $L^{-1}(\{-1,2\})=\{x\in\mathbb R|x<2\}$ (-1 OR 2)
+
 The **empty relation** $\empty$ is a relation on all sets.
 
 The **identity relation** on all sets returns itself.
@@ -672,3 +682,55 @@ The **restriction** of relation $R$ to set $B$ limits a previous relation on a s
 $$R\big|_B=R\cap B^2$$
 
 Graphs are often used to represent relations. A node from $4\to3$ can be represented as $\left<3,4\right>$, much like an adjacency list.
+
+### Reflexivity
+
+A **reflexive** relation $R\subseteq X^2$ is such that every element in $X$ is related to itself by the relation.
+
+$$\forall x\in X,\left<x,x\right>\in R$$
+
+An **irreflexive** relation is such that each element is *not* related to itself.
+
+$$\forall x\in X,\left<x,x\right>\not\in R$$
+
+Reflexivity is determined graphically by checking if the main diagonal of a truth table is true.
+
+!!! example
+    For the reflexive relation $R$, $A=\{1,2\},R=\{\left<1,1\right>,\left<2,2\right>\}$:
+    
+    |$A\times A$ | 1 | 2 |
+    | --- | --- | --- |
+    | 1 | T | F |
+    | 2 | F | T |
+
+!!! warning
+    $\empty$ is often vacuously true for most conditions.
+
+If $R$ is a **non-empty** relation on a **non-empty** set $X$, $R$ cannot be both reflexive and irreflexive.
+
+### Symmetry
+
+A **symmetric** relation $R\subseteq X^2$ is such that every relation goes both ways.
+
+$$\forall x,y\in X^2,\left<x,y\right>\in R\iff\left<y,x\right>\in R$$
+
+An **asymmetric** relation is such that **no** relation goes both ways.
+
+$$\forall x,y\in X^2,\left<x,y\right>\in R\implies\left<y,x\right>\not\in R$$
+
+An **antisymmetric** relation is such that **no** relation goes both ways, *except* if compared to itself, and that the relation relates identical items.
+
+$$\forall x,y\in X^2,\left<x,y\right>\in R\wedge\left<y,x\right>\in R\implies x=y$$
+
+Where $x,y,z$ are elements in $X$, and $p,q,r$ are arbitrary proposition results (true/false):
+
+- Symmetric relations must be symmetrical across the main diagonal of a truth table.
+
+| $X^2$ | $x$ | $y$ | $z$ |
+| --- | --- | --- | --- |
+| $x$ | ? | $p$ | $q$ |
+| $y$ | $\neg p$ | ? | $r$ |
+| $z$ | $\neg q$ | $\neg r$ | ? |
+
+- Asymmetric relations must be oppositely symmetrical across the main diagonal. The main diagonal also must be false.
+- Antisymmetric relations must be false only if there is a true.