ece108: start relations

docs/1b/ece108.md

@@ -622,4 +622,53 @@ A set is **finite** if it is empty or it is mappable to a subset of the natural
 
 $$\exists n\in\mathbb N,\exists f\text{ is bijective}, f:S\to \mathbb N_n,|s|=n$$
 
+### Uncountable sets
 
+The cardinality of countable sets is relative to the cardinality of the set of **natural numbers**.
+
+$$|\mathbb N|=\aleph_0$$
+
+By Contor's theorem, the powerset of the natural numbers must have a larger cardinality than the set of natural numbers.
+
+$$|X|=\aleph_0\implies|\mathcal P(X)|=2^{\aleph_0}>\aleph_0$$
+
+The following can be taken for granted:
+
+- $|\mathbb R|>|\mathbb N|$
+- $|\mathcal P(\mathbb N)|>|\mathbb N|$
+- $|\mathcal P(\mathbb N)|=|\mathbb R|$
+
+## Relations
+
+A **binary relation** $R$ from sets $A$ to $B$ must be a subset of the two. A relation from $A$ to $A$ can be written as $R\subseteq A^2$.
+
+$$R\subseteq A\times B$$.
+
+!!! example
+    - $\forall x,y\in A,B,x<y$ is a subset. $<$ is a binary relation.
+
+For $R\subseteq X\times Y$:
+
+- $\text{dom}(R)=\{x\in X|\exists y\in Y,xRy\}$
+- $\text{cod}(R)=Y$
+- $\text{rang}(R)=\{y\in Y|\exists x\in X,xRy\}$
+- The **image** of $X_1\subseteq X$ under $R$: $R(X_1)=\{y\in Y|\exists x\in X_1xRy\}$
+- The **pre-image** is: $R^{-1}(Y_1)=\{x\in X|\exists y\in Y_1,xRy\}$
+
+Relations are trivially proven to be relations through subset analysis.
+
+The **empty relation** $\empty$ is a relation on all sets.
+
+The **identity relation** on all sets returns itself.
+
+$$E=\{\left<a,a\right>|a\in A\}$$
+
+The **universal relation** relates each element in the first set to every element to the second set.
+
+$$U=A^2$$
+
+The **restriction** of relation $R$ to set $B$ limits a previous relation on a superset $A$ such that $B\subseteq A$.
+
+$$R\big|_B=R\cap B^2$$
+
+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.