---
type: theoretical
---

## Sets

A **set** is an unordered collection of unique objects (elements) with no duplicates.

### Notation
- $x \in A$: "x is an element of set A."
- $x \notin A$: "x is not an element of set A."

### Intervals of $\mathbb{R}$
An **interval** is a subset of $\mathbb{R}$ (the real numbers) defined by two endpoints $a$ and $b$. Any real number between $a$ and $b$ belongs to the interval.

### Cardinality
The **cardinality** of a set $S$ is the number of elements it contains, denoted $|S|$.
- **Finite Set**: A set with a finite number of elements. Example: $S = \{a, b, c\}$, $|S| = 3$.
- **Infinite Set**: A set that is not finite. Example: $\mathbb{N}, \mathbb{Z}, \mathbb{R}$.
- **Power Set**: For a finite set $S$, the cardinality of its power set $P(S)$ (set of all subsets of $S$) is $2^{|S|}$.

**Example**: For $S = \{a, b, c\}$, $P(S)$ contains $8 = 2^3$ subsets.

---

## Set Operations

Let $U$ be the universal set, and let $A$ and $B$ be subsets of $U$:

- **Union**: $A \cup B = \{x \mid x \in A \lor x \in B\}$
- **Intersection**: $A \cap B = \{x \mid x \in A \land x \in B\}$
- **Set Difference**: $A \setminus B = \{x \mid x \in A \land x \notin B\}$
- **Complement**: $\bar{A} = U \setminus A$, the set of all elements not in $A$.

---

## Proof Styles for Set Properties

### Annotated Linear Proof (ALP)
A **chain-of-equivalences proof** involves showing the equivalence of two sets step by step.

**Example**: Prove $A \cup (B \cup C) = (A \cup B) \cup C$.

- Start from $x \in A \cup (B \cup C)$ and simplify:
  $$
  x \in A \cup (B \cup C) \iff x \in A \lor (x \in B \lor x \in C) \iff (x \in A \lor x \in B) \lor x \in C \iff x \in (A \cup B) \cup C.
  $$

- Conclude that the left-hand side equals the right-hand side.

---

## Sequences

A **sequence** is an ordered list of objects where repetition is allowed, and the order matters.

### Examples
1. Finite sequence: $1, 2, 3, 5, 7$ (length 5).
2. Infinite sequence: $0, 2, 4, 6, \ldots$ (continues indefinitely).
3. A sequence differs from a set: $1, 2, 3$ is not the same as $3, 2, 1$.

### Specifying Sequences
1. **Recursive Definition**:
   - Example: $s_0 = 3$, $s_n = 2 \cdot s_{n-1} + 7$, defines $3, 13, 33, \ldots$.
2. **Explicit Definition**:
   - Example: $s_n = n^2$ defines $0, 1, 4, 9, 16, \ldots$.

---

## Characteristic Function

The **characteristic function** of a set $A \subseteq U$ is a function $f_A : U \to \{0, 1\}$:
$$
f_A(x) =
\begin{cases} 
1, & \text{if } x \in A, \\
0, & \text{if } x \notin A.
\end{cases}
$$

### Purpose
Characteristic functions represent sets mathematically and are useful for computations.

**Example**:
- $U = \{a, b, c, d, e\}$, $A = \{b, d\}$.
- Representation of $U$: $(1, 1, 1, 1, 1)$.
- Representation of $A$: $(0, 1, 0, 1, 0)$.

---

## Strings and Languages

### Formal Languages
1. **Alphabet ($\Sigma$)**: A non-empty set of symbols (letters).
2. **Word (String)**: A finite sequence of symbols from $\Sigma$.
   - Example: $\Sigma = \{a, b\}$, string: $babab$.
3. **Empty String ($\epsilon$)**: A string with no symbols ($|\epsilon| = 0$).
4. **Language ($L$)**: A subset of $\Sigma^*$, the set of all possible strings over $\Sigma$.
Regex. Used for [Pattern matching](Pattern%20matching.md)
### Regular Sets and Expressions
1. $\alpha \cdot \beta$ (concatenation): Combine two sets of strings.
2. $\alpha | \beta$ (union): Combine strings from either set.
3. $\alpha^*$ (Kleene star): All strings formed by repeatedly concatenating elements of $\alpha$.

**Example**:
- Let $L$ correspond to $(a | b)^*(c | \epsilon)$.
  - True/False:
    - $abbac \in L$: True.
    - $abccc \in L$: False.
    - $abaa \in L$: False.


---

## Integers

### Primes
- **Prime Number**: A positive integer greater than 1 with only two divisors: 1 and itself.
  - Examples: $2, 3, 5, 7$.
- **Euclid's Theorem**: There are infinitely many primes.

**Proof Sketch**:
1. Assume a finite list of primes $p_1, p_2, \ldots, p_n$.
2. Let $q = (p_1 \cdot p_2 \cdot \ldots \cdot p_n) + 1$.
3. If $q$ is prime, it’s not in the list.
4. If $q$ is not prime, it must be divisible by some prime not in the list.

### Prime Factorization
Every integer $n > 1$ can be uniquely expressed as a product of primes:
$$
n = p_1^{k_1} \cdot p_2^{k_2} \cdot \ldots \cdot p_s^{k_s}.
$$

**Examples**:
1. $60 = 2^2 \cdot 3 \cdot 5$.
2. $168 = 2^3 \cdot 3 \cdot 7$.

---

## Matrices and Boolean Operations

### Boolean Operations
- **OR ($\lor$)**: $a \lor b = \max(a, b)$.
- **AND ($\land$)**: $a \land b = \min(a, b)$.

**Example**: For $a, b \in \{0, 1\}$:
- $1 \lor 0 = 1$.
- $1 \land 0 = 0$.

---

## Mathematical Structures

### Arity and Notation
- **Arity**: The number of arguments an operation takes.
  - **Unary**: Takes one argument (e.g., complement of a set).
  - **Binary**: Takes two arguments (e.g., addition: $x + y$).
  - **Nullary**: Takes no arguments (e.g., constant: $1$).

**Prefix/Infix/Postfix Notation**:
- **Prefix**: Operator first (e.g., $+ x y$).
- **Infix**: Operator between arguments (e.g., $x + y$).
- **Postfix**: Operator last (e.g., $x y +$).