math 101

### Summary of MTH 107: Sets and Numbers

#### 1. Sets

- Definition: A set is a collection of well-defined, distinct objects (elements). Sets are denoted by capital letters.

- Types:

- Finite Set: Has a countable number of elements (e.g., \( A = \{1, 2, 3\} \)).

- Infinite Set: Has uncountable elements (e.g., \( B = \{x \mid x \text{ is an even number}\} \)).

- Empty Set: Contains no elements, denoted by \( \emptyset \) or \(\{\}\).

- Operations:

- Union (\( A \cup B \)): Elements in either \( A \) or \( B \).

- Intersection (\( A \cap B \)): Elements common to both \( A \) and \( B \).

- Complement (\( A' \)): Elements not in \( A \) but in the universal set.

- Difference (\( A - B \)): Elements in \( A \) but not in \( B \).

- Symmetric Difference: Elements in either \( A \) or \( B \) but not both.

- Theorems:

- De Morgan's Laws: \((A \cup B)' = A' \cap B'\) and \((A \cap B)' = A' \cup B'\).

- Inclusion-Exclusion Principle: For counting elements in unions of sets.

#### 2. Number Systems

- Natural Numbers (\( \mathbb{N} \)): Counting numbers \( \{1, 2, 3, \ldots\} \). Closed under addition and multiplication.

- Integers (\( \mathbb{Z} \)): Includes naturals, zero, and negatives \( \{\ldots, -2, -1, 0, 1, 2, \ldots\} \). Closed under addition, subtraction, and multiplication.

- Rational Numbers (\( \mathbb{Q} \)): Numbers expressible as \( \frac{a}{b} \), where \( a, b \in \mathbb{Z} \) and \( b \neq 0 \). Closed under all arithmetic operations except division by zero.

- Real Numbers (\( \mathbb{R} \)): Includes rationals and irrationals (e.g., \( \sqrt{2}, \pi \)). Represents all points on the number line.

- Complex Numbers (\( \mathbb{C} \)): Numbers of the form \( z = x + iy \), where \( x, y \in \mathbb{R} \) and \( i = \sqrt{-1} \). Closed under all operations, including roots of negative numbers.

#### 3. Complex Numbers

- Representation:

- Algebraic Form: \( z = x + iy \).

- Polar Form: \( z = r (\cos \theta + i \sin \theta) \), where \( r = \sqrt{x^2 + y^2} \) (magnitude) and \( \theta = \tan^{-1}(y/x) \) (argument).

- Operations:

- Addition/Subtraction: Combine real and imaginary parts.

- Multiplication/Division: Use algebraic or polar forms for simplification.

- Conjugate: \( \overline{z} = x - iy \).

- De Moivre's Theorem: For \( z = r (\cos \theta + i \sin \theta) \), \( z^n = r^n (\cos n\theta + i \sin n\theta) \).

#### 4. Relations

- Types:

- Reflexive: Every element is related to itself.

- Symmetric: If \( a \) is related to \( b \), then \( b \) is related to \( a \).

- Transitive: If \( a \) is related to \( b \) and \( b \) to \( c \), then \( a \) is related to \( c \).

- Equivalence Relation: Combines reflexive, symmetric, and transitive properties.

#### 5. Applications

- Venn Diagrams: Visual representation of set operations.

- Cartesian Product: \( A \times B = \{(a, b) \mid a \in A, b \in B\} \). Forms the basis for relations.

This course provides foundational knowledge in sets, number systems, and complex numbers, essential for advanced mathematical studies. The material includes definitions, properties, operations, and practical applications like solving quadratic equations and analyzing relations.