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Course Outline: Elementary Set Theory and Related Topics

Elementary Set Theory

  • Subsets:
    • A set A is a subset of a set B (denoted as A ⊆ B) if every element of A is also an element of B.
  • Union:
    • The union of sets A and B (denoted as A ∪ B) is the set of elements that are in A, in B, or in both.
  • Intersection:
    • The intersection of sets A and B (denoted as A ∩ B) is the set of elements that are in both A and B.
  • Complements:
    • The complement of a set A (denoted as A') contains all elements not in A, relative to a universal set U containing all elements under consideration.
  • Venn Diagrams:
    • A visual representation of sets and their relationships, often used to show union, intersection, and complements.

Real Numbers

  • Members of Real Numbers:
    • Integers: Whole numbers that can be positive, negative, or zero (e.g., -3, 0, 7).
    • Rational Numbers: Numbers that can be expressed as the quotient ( \frac{p}{q} ) of two integers, where q is not zero (e.g., 1/2, 3, -4).
    • Irrational Numbers: Numbers that cannot be expressed as a fraction; they have non-repeating, non-terminating decimal expansions (e.g., ( \sqrt{2}, \pi )).

Mathematical Proof Techniques

  • Mathematical Induction:
    • A method of proving statements or formulas are true for all natural numbers. It typically involves a base case and an inductive step.

Sequences and Series

  • Real Sequences:
    • An ordered list of numbers, typically defined by a function f(n) where n is a natural number.
  • Series:
    • The sum of the terms of a sequence, often expressed as ( S = a1 + a2 + a3 + ext{…} + an ).

Quadratic Equations

  • Form:
    • A quadratic equation can be expressed as ( ax^2 + bx + c = 0 ) where a, b, and c are constants, and ( a
      eq 0 ).
  • Solutions:
    • Found using the quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

Binomial Theorem

  • Definition:
    • A formula that provides a way to expand expressions of the form ( (a+b)^n ) as a sum involving terms of the form ( C(n,k)a^{n-k}b^k ), where ( C(n,k) ) is a binomial coefficient.

Complex Numbers

  • Definition:
    • A complex number is of the form ( z = a + bi ) where a and b are real numbers, and i is the imaginary unit satisfying ( i^2 = -1 ).
  • Algebra of Complex Numbers:
    • Includes operations such as addition, subtraction, multiplication, and division, along with properties such as the modulus and argument.

Argand Diagram

  • Definition:
    • A graphical representation of complex numbers on a two-dimensional plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

De Moivre's Theorem

  • Statement:
    • For any complex number expressed in polar form, ( z = r( ext{cos } \theta + i \text{sin } \theta) ), De Moivre's theorem states that ( z^n = r^n( ext{cos } n\theta + i \text{sin } n\theta) ) for any integer n.

nth Roots of Unity

  • Definition:
    • The nth roots of unity are the complex solutions to the equation ( z^n = 1 ), and are given by the formula ( z_k = e^{(2 \pi ik)/n} ) for k = 0, 1, …, n-1.

Circular Measure

  • Definition:
    • A method of measuring angles in radians, where one full rotation corresponds to 2π radians.

Trigonometric Functions

  • Definition:
    • Functions that relate angles to side lengths in right-angled triangles, including sine (sin), cosine (cos), and tangent (tan) functions.
  • Addition and Factor Formulae:
    • Formulas that express the trigonometric functions of sum or difference of angles. For example:
    • ( ext{sin}(a + b) = ext{sin}(a)\text{cos}(b) + \text{cos}(a)\text{sin}(b) )
    • ( ext{cos}(a + b) = \text{cos}(a)\text{cos}(b) - \text{sin}(a)\text{sin}(b) )
    • ( ext{tan}(a + b) = \frac{ ext{tan}(a) + ext{tan}(b)}{1 - ext{tan}(a)\text{tan}(b)}