Numbers, Inequalities, and Absolute Values

Sets

Sets are fundamental collections of objects, called elements. Understanding sets is essential for describing number systems and mathematical relationships.

• Set Notation: Curly brackets list elements, e.g., A = {1, 2, 3, 4}.

• 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): {1, 2, 3, ...}

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

• Rational Numbers (Q): Numbers of the form p/q where p, q ∈ Z, q ≠ 0

• Irrational Numbers: Real numbers not rational, e.g., \sqrt{2}, \pi

• Real Numbers (R): All rational and irrational numbers

Ordering: For a, b ∈ R, b > a means b is to the right of a on the number line.

• a ≤ b means a < b or a = b

• a ≥ b means a > b or a = b

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

• Open interval: (a, b) = {x ∈ R | a < x < b}

• Closed interval: [a, b] = {x ∈ R | a ≤ x ≤ b}

Inequalities

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

• Example: Solve 3x + 1 > 2x:

  • Subtract 2x: x + 1 > 0

  • Subtract 1: x > -1

  • Solution: (-1, \infty)

• Compound Inequalities: Solve 2x < 3x + 1 ≤ 5x - 2:

  • Break into two: 3x + 1 > 2x and 3x + 1 ≤ 5x - 2

  • First: x > -1; Second: x ≥ 3/2

  • Combined (AND): x ≥ 3/2

Absolute Value

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

• Definition: |a| = \begin{cases} a & a ≥ 0 \\ -a & a < 0 \end{cases}

• Key Properties:

  • |a| ≥ 0

  • |a| = 0 if and only if a = 0

  • |ab| = |a||b|

  • |a + b| ≤ |a| + |b| (Triangle Inequality)

  • |a| = \sqrt{a^2}

• Solving Absolute Value Equations:

  • |2x + 4| < 1 ⇒ -1 < 2x + 4 < 1 ⇒ -5 < 2x < -3 ⇒ -5/2 < x < -3/2

Functions

Definition and Basics

A function is a rule that assigns to each element x in a set D (domain) a unique element f(x) in a set Y (codomain).

• Domain: Set of all possible inputs (x-values)

• Range: Set of all possible outputs (f(x)-values)

• Example: f(x) = 1/\sqrt{x-1} has domain (1, \infty)

Graphs of Functions

• The graph of f is the set {(x, f(x)) | x ∈ D}

• Vertical Line Test: A curve is the graph of a function if every vertical line intersects it at most once.

Transformations of Graphs

• Shifting: y = f(x + c) shifts left by c; y = f(x) + c shifts up by c

• Scaling: y = f(cx) compresses horizontally; y = cf(x) stretches vertically

• Reflection: y = f(-x) reflects over y-axis; y = -f(x) reflects over x-axis

Even and Odd Functions

• Even: f(-x) = f(x) for all x in domain (symmetric about y-axis)

• Odd: f(-x) = -f(x) for all x in domain (symmetric about origin)

• Example: f(x) = x^2 is even; f(x) = x^3 is odd

Classification and Combination of Functions

• Polynomial: p(x) = a_n x^n + ... + a_0

• Rational: g(x) = p(x)/q(x) where p, q are polynomials, q(x) ≠ 0

• Algebraic: Built from polynomials using roots and rational operations

• Sum/Product/Quotient: (f + g)(x) = f(x) + g(x), etc.

• Composite: (f \circ g)(x) = f(g(x))

Inverse Functions

• One-to-one (Injective): f(x_1) = f(x_2) ⇒ x_1 = x_2

• Inverse: If f is one-to-one, f^{-1}(y) = x such that f(x) = y

• Graph: The graph of f^{-1} is the reflection of the graph of f across the line y = x

Angles and Trigonometric Functions

Radian Measure

• Definition: The radian measure of an angle is the ratio of arc length s to radius r: \alpha = s/r

• Conversion: Degrees to radians: multiply by \pi/180; Radians to degrees: multiply by 180/\pi

• Arc Length: s = r\alpha (with \alpha in radians)

• Area of Sector: A = (1/2) r^2 \alpha

Trigonometric Functions

• Definitions (unit circle):

  • sin \theta = y/r

  • cos \theta = x/r

  • tan \theta = y/x

  • cosec \theta = r/y

  • sec \theta = r/x

  • cot \theta = x/y

• Special Angles: Know exact values for 0, 30°, 45°, 60°, 90° (see table below)

Trigonometric Identities

• Pythagorean: \sin^2 \theta + \cos^2 \theta = 1

• Double Angle: \sin(2\theta) = 2\sin \theta \cos \theta

• Sum and Difference: \sin(\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi

• Product-to-Sum: \sin \theta \cos \phi = (1/2)[\sin(\theta + \phi) + \sin(\theta - \phi)]

Inverse Trigonometric Functions

• arcsin x: The unique y in [-\pi/2, \pi/2] with \sin y = x

• arccos x: The unique y in [0, \pi] with \cos y = x

• arctan x: The unique y in (-\pi/2, \pi/2) with \tan y = x

Trigonometric Equations

• General Solution for sin θ = x: θ = \arcsin x + 2k\pi or θ = \pi - \arcsin x + 2k\pi, k ∈ \mathbb{Z}

• General Solution for cos θ = x: θ = \arccos x + 2k\pi or θ = -\arccos x + 2k\pi

• General Solution for tan θ = x: θ = \arctan x + k\pi

Polar Coordinates and Graphs

Polar Coordinates

• Definition: A point in the plane is given by (r, θ), where r ≥ 0 is the distance from the origin and θ is the angle from the positive x-axis.

• Conversion:

  • x = r \cos θ

  • y = r \sin θ

  • r = \sqrt{x^2 + y^2}

  • θ = \arctan(y/x) (adjust for quadrant)

Polar Equations and Graphs

• Circle: r = c (center at origin, radius c)

• Line: θ = k (ray at angle k)

• Spiral: r = kθ

• Rose: r = a \cos nθ or r = a \sin nθ (number of petals depends on n)

• Cardioid: r = a + b \sin θ or r = a + b \cos θ

Mathematical Induction

Principle of Mathematical Induction

• To prove a statement for all n ∈ \mathbb{N}:

  1. Show it is true for n = 1 (base case)

  2. Assume true for n = k (inductive hypothesis)

  3. Show it is true for n = k + 1 (inductive step)

Example: Prove 1 + 2 + ... + n = n(n + 1)/2 for all n ∈ \mathbb{N}.

Sigma Notation and Binomial Theorem

Sigma Notation

• Definition: \sum_{j=1}^n a_j = a_1 + a_2 + ... + a_n

• Properties:

  • \sum_{i=r}^n (a_i + b_i) = \sum_{i=r}^n a_i + \sum_{i=r}^n b_i

  • \sum_{i=r}^n k a_i = k \sum_{i=r}^n a_i

Summation Formulas

• Arithmetic Series: \sum_{j=0}^{n-1} (a + jd) = \frac{n}{2}[2a + (n-1)d]

• Geometric Series: \sum_{j=0}^{n-1} ar^j = a \frac{1 - r^n}{1 - r} (for r ≠ 1)

• Sum of First n Integers: \sum_{r=1}^n r = \frac{n(n+1)}{2}

• Sum of Squares: \sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}

• Sum of Cubes: \sum_{r=1}^n r^3 = \frac{n^2(n+1)^2}{4}

Factorials and Binomial Coefficients

• Factorial: n! = n \times (n-1) \times ... \times 1, 0! = 1

• Binomial Coefficient: \binom{n}{k} = \frac{n!}{k!(n-k)!}

• Properties:

  • \binom{n}{0} = \binom{n}{n} = 1

  • \binom{n}{k} = \binom{n}{n-k}

  • \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}

Binomial Theorem

• Statement: (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k

• Pascal's Triangle: The coefficients \binom{n}{k} form Pascal's triangle.

• Example: Expand (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Conic Sections

Quadratic Forms and Canonical Forms

• General Quadratic: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

• Canonical Forms:

  • Parabola: y^2 = 4ax

  • Ellipse: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

  • Hyperbola: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Classification

• Ellipse: Both x^2 and y^2 terms have the same sign

• Hyperbola: x^2 and y^2 terms have opposite signs

• Parabola: Only one variable is squared

Change of Axes

• Translation: Completing the square shifts the origin to simplify the equation

• Rotation: Used to eliminate the xy term; angle \alpha found by \cot 2\alpha = (A - C)/B

Appendix: Mathematical Reasoning

• Statement: An expression that is either true or false

• Implication: "If p then q" (p ⇒ q)

• Converse: "If q then p" (q ⇒ p)

• Contrapositive: "If not q then not p" (¬q ⇒ ¬p)

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

• Proof Methods: Direct, Contradiction, Induction

Additional info: These notes are based on the MATH1034A Algebra lecture manual, covering foundational topics in college algebra and trigonometry, including sets, real numbers, inequalities, functions, trigonometric functions, polar coordinates, mathematical induction, sigma notation, binomial theorem, and conic sections. The structure and examples are designed to support exam preparation and conceptual understanding for college-level algebra students.