Numbers, Inequalities, and Absolute Values

Sets

Sets are fundamental objects in mathematics, representing collections of distinct elements. Understanding set notation and operations is essential for algebraic reasoning.

• Definition: A set is a well-defined collection of objects, called elements.

• Notation: Sets are denoted by curly brackets, e.g., A = {1, 2, 3}.

• Set-builder notation: S = {x | A(x)} means the set of all x for which A(x) is true.

• Subset: A ⊂ B if every element of A is also in B.

• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}.

• Union: A ∪ B = {x | x ∈ A or x ∈ B}.

• Empty set: The set with no elements, denoted ∅.

Example: The set of all continents: C = {Africa, Antarctica, Australia, Asia, Europe, North America, South America}.

Real Numbers

The real numbers encompass several important subsets:

• Natural numbers (N):

• Integers (Z):

• Rational numbers (Q):

• Real numbers (R): All points on the number line, including both rational and irrational numbers.

• Irrational numbers: Real numbers not expressible as a ratio of integers, e.g., .

Intervals: Subsets of the real line, e.g.,

• Open interval:

• Closed interval:

Inequalities

Inequalities describe the relative size of real numbers and are solved using algebraic manipulation.

• Symbols: , , ,

• Solution sets: Expressed in set or interval notation.

Example: Solve :

• Solution:

Absolute Values

The absolute value of a real number measures its distance from zero.

• Definition:

• Key properties:

  • (triangle inequality)

Example: Solve :

Functions

Definition and Basics

A function assigns each element in a domain to a unique element in a codomain.

• Definition: assigns to each .

• Domain: The set of allowable inputs.

• Range: The set of outputs.

Example: has domain .

Graphing Techniques

• Graph: The set of points in the plane.

• Vertical Line Test: A curve is the graph of a function if no vertical line intersects it more than once.

• Piecewise functions: Defined by different formulas on different intervals.

Transformations

• Shifting: (horizontal), (vertical)

• Scaling: (horizontal), (vertical)

• Reflection: (about y-axis), (about x-axis)

Even and Odd Functions

• Even: (symmetric about y-axis)

• Odd: (symmetric about origin)

Example: is even; is odd.

Classification and Combination of Functions

• Polynomial:

• Rational:

• Algebraic: Built from polynomials using roots and rational operations.

Operations:

Composite and Inverse Functions

• Composite:

• Inverse: if and is one-to-one

• Horizontal Line Test: is one-to-one if every horizontal line meets its graph at most once.

Example: has inverse .

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio of arc length to radius :

• Conversion: multiply by ; multiply by

• Arc length: (with in radians)

• Area of sector:

Trigonometric Functions

• Definitions (unit circle):

• Periods: have period ; have period .

Special Angles Table:

Trigonometric Identities

Inverse Trigonometric Functions

• arcsin: is the unique such that

• arccos: is the unique such that

• arctan: is the unique such that

Solving Trigonometric Equations

• General solution for : or ,

• General solution for : or

• General solution for :

Polar Coordinates

Definition and Conversion

• Polar coordinates: where is the distance from the origin, is the angle from the positive x-axis.

• Conversion to Cartesian: ,

• Conversion to Polar: , (quadrant check required)

Polar Equations and Graphs

• Circle:

• Line through origin:

• Spiral:

• Rose: or

• Cardioid: or

Symmetry tests:

• x-axis: Replace with

• y-axis: Replace with

• Origin: Replace with

Mathematical Induction

Principle of Mathematical Induction

• To prove a statement for all :

  1. Show is true (base case).

  2. Assume is true; show is true (inductive step).

Example: Prove for all .

Sigma Notation and Binomial Theorem

Sigma Notation

• Definition:

• Properties:

Summation Formulas

Factorials and Binomial Coefficients

• Factorial: ,

• Binomial coefficient:

Binomial Theorem

• Statement:

• Pascals Triangle: The coefficients form Pascal's triangle.

Example: Expand

Conic Sections

Quadratic Forms and Canonical Forms

• General quadratic:

• Canonical forms:

  • Parabola:

  • Ellipse:

  • Hyperbola:

Classification:

• Parabola: One variable squared, one linear

• Ellipse: Both variables squared, same sign

• Hyperbola: Both variables squared, opposite signs

Transformations: Completing the square and rotation of axes can reduce any non-degenerate quadratic to canonical form.

Appendix: Basic Mathematical Notions

• Statement: An expression that is either true or false.

• Theorem: A true statement proved from axioms and definitions.

• Axiom: A statement assumed to be true without proof.

• Implication: "If p then q" ()

• Equivalence: "p if and only if q" ()

• Contrapositive: The contrapositive of is

• Proof by contradiction: Assume the negation and derive a contradiction.

Intervals:

Additional info: These notes are structured to provide a comprehensive overview of foundational topics in college algebra and trigonometry, including definitions, properties, examples, and key formulas. For more advanced proofs and applications, consult the referenced textbook sections.