Introduction to Set Theory

What is a Set?

• A set is a collection of objects, known as elements.
• Elements can be physical objects, thoughts, ideas, or mathematical objects.
• Sets package objects with similar properties meaningfully.

Characteristics of Sets

• Unambiguous Membership: Clear distinction if an object is in a set.
• Claims and Truth: Can make and assess true/false claims about sets.
• Notation: Sets written with curly brackets {}; elements separated by commas.

Naming and Symbols

• Sets can be named for ease, e.g., A = {1, 2, 3}.
• Membership is symbolized by ∈ (e.g., 1 ∈ A) and non-membership by ∉.

Set Builder Notation

• Describes sets without listing all elements.
• Example: P = {p | p is prime} using the predicate for membership.

Declaring Initial Sets

• Important to declare initial sets for context.
• Example predicates: p ∈ ℕ such that p < 5 or r ∈ ℝ such that r < 5.

Set Equality

• Sets are equal if they contain the same elements.
• Order and repetition of elements do not affect set equality.

Cardinality

• Size of a set; number of elements it contains.
• Denoted by vertical lines |A|.
• Infinite sets can use ∞.

Subsets and Proper Subsets

• Subset: All elements of one set are in another.
• Proper Subset: Subset not equal to the other set.
• Symbolized differently in contexts.
• Transitivity: If A ⊆ B and B ⊆ C, then A ⊆ C.

Empty Set

• Contains no elements; unique and a subset of any set.

Operations on Sets

Union and Intersection

• Union (A ∪ B): All elements in either set.
• Intersection (A ∩ B): Elements common to both sets.
• Properties: Order of operation does not matter, and sets can be combined in any order.
• Identity for Union: A ∪ ∅ = A
• Identity for Intersection: A ∩ ∅ = ∅

Set Difference

• Difference (A \ B): Elements in A not in B.
• Complement: Special difference between a set and its universal context.

De Morgan's Laws

• Complement of Union: (A ∪ B)' = A' ∩ B'
• Complement of Intersection: (A ∩ B)' = A' ∪ B'

Power Set

• Contains all subsets of a given set.

Indexed Families of Sets

• Sets indexed by numbers, often written with subscripts.

Advanced Topics

Russell's Paradox

• Paradox arises from a set that contains itself, leading to contradictions.
• Highlights the need for axiomatic set theory.

Axiomatic Set Theory

• Provides rigorous definitions and axioms for sets to avoid paradoxes.

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This lecture covered the foundational aspects of set theory, including definitions, notations, operations, and concepts such as subsets, complements, and paradoxes. Understanding these basics is crucial for further exploration in mathematical set theory and its applications.