ece108: add functions intro

docs/1b/ece108.md

@@ -407,7 +407,7 @@ $$[a,b)=\{x\in\mathcal U|a\leq x\wedge x<b\}$$
 
 $\empty$ is any impossible interval.
 
-## Ordered pairs
+### Ordered pairs
 
 !!! definition
     - A **binary relation** on two sets $A, B$ is a subset of their Cartesian product.
@@ -426,3 +426,93 @@ It is effectively the cross product, so is not commutative, although distributin
 The **n-Cartesian product** of $n$ sets expands the Cartesian product.
 
 $$A\times B\times\dots\times Z=\{\left<a, b,\dots z\right>|a\in A, b\in B,\dots,z\in Z\}$$
+
+### Powersets
+
+!!! definition
+    - An **index set** $I$ is the set containing all relevant indices.
+
+A **partition** of a set $S$ is a set of **disjoint** sets that create the original set when unioned.
+
+$$S=\bigcup_{i\in I}A_i$$
+
+!!! example
+    $\{\{1\},\{2,3\},\{4,\dots\}\}$ is a partition of $\mathbb N$.
+    
+A **powerset** of a set $A$ is the set of all possible subsets of that set.
+
+$$\mathcal P(A)=\{X|X\subseteq A\}$$
+
+The empty set is the subset of every set so is part of each powerset. The number of elements in a subset is equal to the the number of elements in the original set as a power of two.
+
+$$\dim(\mathcal P(A))=2^{\dim(A)}$$
+
+!!! example
+    - $\mathcal P(\empty)=\empty$
+    - $\mathcal P(\{1,2\})=\{\empty, \{1\}, \{2\}, \{1, 2\}\}$
+
+By definition, any subset is an element in the powerset.
+
+$$A\subseteq B\equiv A\in\mathcal P(B)$$
+
+- $\empty\in\mathcal P(A)$
+- $A\in\mathcal P(A)$
+- $A\subseteq B\implies (\mathcal P(A)\subseteq \mathcal P(B))$
+- $A\in C\implies (C-A\subseteq C)$
+
+!!! example
+    To prove $A\subseteq B\implies \mathcalP(A)\subseteq \mathcal P(B)$:
+    
+    **Proof:** Let $A\subseteq B$ and $X\in\mathcal P(A)$. By definition, since $X\in\mathcal P(A), X\subseteq A$. Since $A\subseteq B$, it follows that $X\subseteq B$. Thus by the definition of the powerset, $X\in\mathcal P(B)$.
+    
+## Functions
+
+!!! definition
+    - A **surjective** function has an equal codomain and range.
+
+A **function** a relation between two sets $f:X\to Y$ such that each $x\in X$ **maps to** a unique $f(x)\in Y$.
+
+$$
+\begin{align*}
+f:\ &X\to Y \\
+&x\longmapsto f(x)
+\end{align*}
+$$
+
+!!! example
+    Sample function with multiple cases and indices:
+    
+    $$
+    \begin{align*}
+    f:\ &X\to Y \\
+    &x_i\longmapsto \begin{cases}
+    y_1 & i\in\{1,2\} \\
+    y_3 & i\in\{3,4,5\}
+    \end{cases}
+    \end{align*}
+    $$
+
+The **domain** $\text{dom}(f)$ is the input set.
+
+$$X=\text{dom}(f)$$
+
+The **codomain** $\text{cod}(f)$ is the output set.
+
+$$Y=\text{cod}(f)$$
+
+The **range** $\text{rang}(f)$ is the subset of $Y$ that is actually mapped to by the domain.
+
+$$
+\begin{align*}
+\text{rang}(f)&=\{y\in Y|\exists x\in X,y=f(x)\} \\
+&=\{f(x)|x\in X\}
+\end{align*}
+$$
+
+The **pre-image** is the subset of the domain that maps to a specific subset $B$ of the codomain.
+
+$$\text{preimage}(f)=\{x\in X|\exists y\in B,y=f(x)\}$$
+
+The **image** is the subset of the codomain that is mapped by a specific subset $A$ of the domain.
+
+$$\text{image}(f)=\{f(x)|\exists x\in A\}$$