Introduction to University Mathematics

Overview of the Course

• Introduction to a more formal and rigorous way of presenting mathematical arguments compared to school.
• Topics covered:
  • Basic set theory
  • Logic and functions
  • Examples of mathematical proof and problem solving.
• Aim: to introduce notation and mathematical language necessary for advanced mathematics.

Natural Numbers

Definition

• Set of natural numbers (N) includes non-negative integers: 0, 1, 2, ...
• Notation: N = {0, 1, 2, ...}
• Ambiguity regarding inclusion of 0:
  • Zero is included for this course, but not universally accepted.

Properties of Natural Numbers

• Operations: Addition and multiplication resulting in natural numbers.
  • Notation: If M, N ∈ N, then M + N and M × N ∈ N.
  • M × N can be written as MN.
• Identities:
  • Additive identity: 0 (N + 0 = N).
  • Multiplicative identity: 1 (N × 1 = N).
• Ordering: Based on the definition of less than or equal to (M ≤ N):
  • M ≤ N ↔ there exists K ∈ N such that M + K = N.

Mathematical Induction

Principle of Mathematical Induction

• To prove P(N) for all natural numbers:
  1. Prove P(0).
  2. Prove if P(N) is true, then P(N + 1) is true.
• Visual analogy: Domino Effect.

Example: Sum of Integers

• Proposition: Sum from 0 to N = (1/2)N(N + 1).
• Proof involves:
  1. Proving base case P(0) is true.
  2. Assuming P(N) is true, then showing P(N + 1).

Strong Induction Variant

• Can use statements P(0), P(1), ..., P(N) to prove P(N + 1).
• Example: Every natural number greater than 1 is the product of primes.

Properties of Natural Numbers

Associative Property of Addition

• Proposition: (X + Y) + Z = X + (Y + Z).
• Inductive proof on Z:
  1. Base case (Z = 0).
  2. Inductive step (Z = N, then Z = N + 1).

Well-Ordering Property

• Proposition: Every non-empty subset of natural numbers has a least element.
• Proof by contradiction combined with induction.
  1. Assume a non-empty subset S has no least element.
  2. Define complement set S* and show it leads to contradiction.

Summary

• Discussed defining natural numbers, operations, and ordering.
• Explored induction and its applications.
• Proved properties like commutativity of addition and well-ordering property.