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.

• Set: A well-defined collection of objects, called elements.

• Notation: Curly brackets list elements, e.g., {1, 2, 3}.

• Set-builder notation: {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.

Example: The set of all natural numbers between 18 and 956: {x ∈ N | 18 < x < 956}

Real Numbers

The real number system is built from several subsets:

• Natural numbers (N): {1, 2, 3, ...}

• Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Rational numbers (Q): Numbers expressible as , where , .

• Irrational numbers: Real numbers not rational, e.g., , .

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

Ordering: For , if is to the right of on the number line.

• means or .

• means or .

Intervals

Intervals are subsets of the real line, often used to describe solution sets.

Inequalities

Solving inequalities involves finding all real numbers that satisfy a given relation.

• Example: Solve :

  • Solution:

• Compound inequalities:

  • Break into two: and

  • Find intersection of solution sets.

Absolute Value

The absolute value of is its distance from zero on the real number line.

• Definition:

• Key properties:

  • for all

  • (triangle inequality)

  • (for )

  • or (for )

Example: Solve :

Functions

Definition and Basics

A function from set to set assigns to each a unique .

• Domain: The set of all possible inputs.

• Range: The set of all possible outputs.

• Real-valued function: , .

Example: has domain (since and denominator ).

Graphing Functions

• Graph: The set of points for .

• 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 parts of the domain.

Transformations of Graphs

• Shifting: (horizontal), (vertical)

• Scaling: (horizontal), (vertical)

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

Even and Odd Functions

• Even: for all in domain. Graph is symmetric about -axis.

• Odd: for all in domain. Graph is symmetric about the origin.

• Example: is even; is odd.

Classification and Combination of Functions

• Polynomial:

• Rational: ,

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

• Operations: , , (where )

• Composite:

Inverse Functions

• One-to-one (injective):

• Inverse: If is one-to-one, where

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

• Graph of inverse: Reflection of 's graph across the line .

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio , where is arc length and is radius.

• Arc length: (with in radians)

• Conversion: $1= \frac{180}{\pi} degree radians

Area of a Sector

• Formula: (with in radians)

Trigonometric Functions

Special Angles Table

Trigonometric Identities

Inverse Trigonometric Functions

• arcsin: is the unique with

• arccos: is the unique with

• arctan: is the unique with

Trigonometric Equations

• General solution for (): or ,

• General solution for (): or

• General solution for :

Polar Coordinates

• Definition: where is the distance from the origin, is the angle from the positive -axis.

• Conversion: , ; ,

• Polar equations: Equations in and describe curves in the plane (e.g., is a circle of radius 2).

Expressions of the Form

• Can be rewritten as , where ,

• Example:

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).

• If both steps hold, is true for all .

Example: Prove for all by induction.

Sigma Notation and Binomial Theorem

Sigma Notation

• Definition:

• Properties:

Summation Formulas

Factorials and Binomial Coefficients

• Factorial: ,

• Binomial coefficient:

• Properties:

Binomial Theorem

• For ,

• Pascal's Triangle: The coefficients form the entries of Pascal's triangle.

• Example:

Conic Sections

Quadratic Forms and Canonical Forms

• General quadratic form:

• Canonical forms:

  • Parabola:

  • Ellipse:

  • Hyperbola:

Classification

• Parabola: One squared term, e.g.,

• Ellipse: Both and terms, same sign

• Hyperbola: Both and terms, opposite signs

Change of Axes

• Translation: Completing the square to shift the origin

• Rotation: Removing the term by rotating axes through angle where

Summary Table: Conic Sections

Appendix: Mathematical Reasoning

Logic and Proof

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

• Implication: "If p then q" ()

• Converse:

• Contrapositive: (logically equivalent to the original implication)

• Equivalence: (both and are true)

• Direct proof: Prove directly.

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

• Counterexample: A single example disproving a statement.

Properties of Real Numbers

• Associative, commutative, distributive laws for addition and multiplication

• Order properties: or ; if $a \leq b$ and $b \leq a$, then

• Intervals: See earlier table for notation and types

Additional info: This summary covers all major topics from the provided lecture manual, including definitions, properties, examples, and key formulas, and is structured for exam preparation in a college algebra course.