Lecture 1: Introduction to Sets in Mathematics

Key Symbols for Sets

• Curly Brackets: { and } indicate the beginning and end of a set.
• Comma ,: Used to separate elements within a set.

Basic Definitions

• Set: A collection of distinct objects, considered as an object in its own right.
• Element of a Set: An object that is a member of a set.
  • Example: apple is an element of set S = {apple, orange}.
• Not an Element: Denoted by ∉.
  • Example: pear ∉ S.

Properties of Sets

• Order & Multiplicity: The order of elements and the number of times an element is listed do not matter.
  • Example: {apple, orange} is the same as {orange, apple} and {apple, orange, apple}.

Constructing Sets

• List Form: Directly listing all elements, e.g., S = {1, 3, 5}.
• Predicate Form: Describes properties elements must satisfy.
  • Example: Set A can be constructed as {x ∈ ℕ | x = 2n + 1, x ≤ 6} resulting in {1, 3, 5}.
  • Another example with quadratic equation: B = {z ∈ ℤ | z² - 3z + 2 = 0} yields {1, 2}.

Special Sets

• Empty Set: Denoted by {} or ∅.
  • Note: {} is not equal to {{}} (the latter is a set containing the empty set).

Examples of Sets Used in Mathematics

• Natural Numbers (ℕ): {1, 2, 3, ...}
  • Sometimes includes zero: {0, 1, 2, ...}
• Integers (ℤ): {..., -2, -1, 0, 1, 2, ...}
• Rational Numbers (ℚ): {a/b | a ∈ ℤ, b ∈ ℕ, gcd(a, b) = 1}
• Real Numbers (ℝ): Numbers with a decimal expansion, e.g., z.a₁a₂a₃...
  • Handles duplicates like 0.999... = 1.0
• Complex Numbers (ℂ): {a + ib | a ∈ ℝ, b ∈ ℝ}

Historical Note

• Introduction of Zero: Zero was introduced by Brahmagupta, an Indian mathematician.

This lecture provides an overview of basic set theory concepts and introduces key types of numbers used in mathematics.