2.1 Set Definitions and Notations

Set theory forms the foundation of mathematical reasoning. It introduces the concept of collections and their relationships. Understanding sets is crucial for grasping more complex mathematical ideas and structures.

This section covers the basics of set definitions and notations. We'll learn about elements, subsets, and how to represent sets using different methods. These fundamentals will help us explore set operations and properties later in the chapter.

Set Definitions

Fundamental Set Concepts

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• Set represents a collection of distinct objects or elements grouped together
• Element refers to any individual object within a set, denoted by the symbol ∈
• Subset consists of a collection of elements from another set, symbolized by ⊆
• Empty set contains no elements, represented by ∅ or {}
• Universal set encompasses all elements under consideration for a particular context, often denoted by U

Set Representation and Relationships

• Sets commonly depicted using capital letters (A, B, C)
• Elements written as lowercase letters or numbers (a, b, 1, 2)
• Subset relationship expressed as A ⊆ B, meaning every element in A is also in B
• Proper subset (A ⊂ B) occurs when A is a subset of B, but A ≠ B
• Empty set serves as a subset of every set, including itself

Set Cardinality and Properties

• Cardinality measures the number of elements in a set, denoted by |A|
• Singleton set contains exactly one element
• Power set includes all possible subsets of a given set, represented by P(A)
• Disjoint sets have no common elements (A ∩ B = ∅)
• Equal sets contain the same elements (A = B if and only if A ⊆ B and B ⊆ A)

Set Notations

Set-Builder Notation

• Describes sets using a rule or condition elements must satisfy
• General form: {x | P(x)}, read as "the set of all x such that P(x) is true"
• Allows for concise representation of infinite sets
• Utilizes mathematical symbols and logical connectives
• Can incorporate multiple conditions using conjunctions (and) or disjunctions (or)

Roster Notation

• Lists all elements of a set within curly braces {}
• Separates elements using commas
• Employs ellipsis (...) to indicate continuation of a pattern
• Suitable for finite sets or infinite sets with recognizable patterns
• Can combine with set-builder notation for complex sets

Comparing Set Notations

• Set-builder notation excels at describing sets with specific properties
• Roster notation proves more intuitive for small, finite sets
• Some sets can be represented equally well using either notation
• Choosing between notations depends on set characteristics and context
• Proficiency in both notations enhances mathematical communication skills

Set Types

Finite Sets

• Contain a countable number of elements
• Can be listed completely using roster notation
• Cardinality represented by a non-negative integer
• Include empty set (cardinality 0) and singleton sets (cardinality 1)
• Often encountered in practical applications (days of the week)

Infinite Sets

• Contain an unlimited number of elements
• Cannot be fully listed using roster notation
• Described using set-builder notation or patterns with ellipsis
• Divided into countably infinite (ℕ, ℤ) and uncountably infinite (ℝ) sets
• Require special techniques for comparing sizes (cardinality)

Set Type Characteristics

• Finite sets always have a maximum and minimum element (if non-empty)
• Infinite sets may or may not have maximum or minimum elements
• Bounded sets have both upper and lower bounds (finite sets are always bounded)
• Unbounded sets extend infinitely in at least one direction
• Some sets can be both countably infinite and unbounded (ℤ)