ece108: add set intro

docs/1b/ece108.md

@@ -294,3 +294,63 @@ Induction can be applied to the whole set of integers by proving the following:
 
 Alternatively, some steps can be skipped in **strong induction** by proving that if for $k\in\mathbb N$, $P(i)$ holds for all $i\leq k$, so $P(k+1)$ holds. In other words, by assuming that the statement is true for all values before $k$. If strong induction is true, regular induction must also be true, but not vice versa.
     
+## Sets
+
+!!! definition
+    - A **set** is an unordered collection of distinct objects.
+    - An **element/member** of a set is an object in that set.
+    - A **multiset** is an unordered collection of objects.
+
+Sets are expressed with curly brackets:
+
+$$\{s_1, s_2,\dots\}$$
+
+Numbers are defined as sets of recursively empty sets:
+
+$$
+\begin{align*}
+0&:=\empty \\
+1&:=\{\empty\} \\
+2&:=\{\empty,\{\empty\}\}
+\end{align*}
+$$
+
+### Special sets
+
+- $\mathbb N$ is the set of **natural numbers** $\{1, 2, 3,\dots\}$
+- $\mathbb W$ is the set of **whole numbers** $\{0, 1, 2,\dots\}$
+- $\mathbb Z$ is the set of **integers** $\{\dots, -1, 0, 1, \dots\}$
+- $\mathbb Z^+_0$ is the set of **positive integers, including zero** — these modifiers can be applied to the set of negative integers and real numbers as well
+- $2\mathbb Z$ is the set of **even integers**
+- $2\mathbb Z + 1$ is the set of **odd integers**
+- $\mathbb Q$ is the set of **rational numbers**
+- $\mathbb R$ is the set of **real numbers**
+- $\empty$ or $\{\}$ is the **empty set** with no elements
+
+### Set builder notation
+
+!!! definition
+    - The **domain of discourse** is the context of the current problem, which may limit the universal set (e.g., if only integers are discussed, the domain is integers only)
+
+$x$ is an element if $x$ is in $\mathcal U$ and $P(x)$ is true.
+
+$$\{x\in\mathcal U|P(x)\}$$
+
+!!! example
+    All even numbers: $A=\{n\in\mathbb Z,\exists k\in\mathbb Z,n=2k\}$
+
+$f(x)$ is an element if $x$ is in $\mathcal U$, and $P(x)$ is true:
+
+$$\{f(x)|\underbrace{x\in\mathcal U, P(x)}_\text{swappable, omittable}\}$$
+
+!!! example
+    - All even numbers: $A=\{2k|k\in\mathbb Z\}$
+    - All rational numbers: $\mathbb Q=\{\frac a b | a,b\in\mathbb Z,b\neq 0\}$
+
+The **complement** of a set is the set containing every element **not** in the set.
+
+$$\overline S$$
+
+The **universal set** is the set containing everything, and is the complement of the empty set.
+
+$$\mathcal U=\overline\empty$$

docs/1b/math119.md

@@ -244,3 +244,4 @@ Sample tree diagram:
 
 !!! warning
     If the function only depends on one variable, $\frac{d}{dx}$ is used. Multivariable functions must use $\frac{\partial}{\partial x}$ to treat the other variables as constant.
+