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Fundamentals of Set Theory Explained
Nov 16, 2024
Lecture Notes on Set Theory
Introduction to Sets
A set is a collection of objects called elements.
Elements can be physical objects, thoughts, ideas, or mathematical objects.
Sets help in organizing objects with shared properties meaningfully.
Definition and Properties of Sets
Example of sets
: Set of triangles.
Clear criteria to determine membership.
Claims about sets can be evaluated as true or false.
Example: Elements of triangles have three sides; the sum of internal angles is not always 360 degrees.
Sets are represented using curly brackets: {1, 2, 3}.
Sets can be named for easier reference (e.g., let A = {1, 2, 3}).
Membership notation
:
1 ∈ A (1 is in A)
4 ∉ A (4 is not in A)
Set Builder Notation
Example
: Set of prime numbers can be expressed as P = {p | p is prime}.
A variable must satisfy a certain criterion (predicate) to belong to the set.
Explicitly declare starting sets before the predicate (e.g., the natural numbers).
Equality and Subsets
Two sets are equal if they contain the same elements, regardless of order or repetition.
Cardinality
: The size of a set, denoted by ||A||.
A set A is a subset of B if all elements of A are in B.
Notation: A ⊆ B.
Example: If A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}, then A is a subset of B.
A proper subset (A ⊂ B) means A is a subset of B but not equal to B.
All sets are subsets of themselves.
The Empty Set
Definition
: A set with no elements, denoted by {} or ∅.
The empty set is a subset of any set and is unique.
Venn Diagrams
Union of Sets
: A ∪ B = {x | x ∈ A or x ∈ B}.
Intersection of Sets
: A ∩ B = {x | x ∈ A and x ∈ B}.
Example: Let A = {0, 1} and B = {1, 2, 3}.
Union: A ∪ B = {0, 1, 2, 3}.
Intersection: A ∩ B = {1}.
Properties of unions and intersections:
A ∪ ∅ = A and A ∩ ∅ = ∅.
A ∪ A = A and A ∩ A = A.
Cardinality of Union and Intersection
Identity
: |A ∪ B| = |A| + |B| - |A ∩ B|.
This leads to an inequality: |A ∪ B| ≤ |A| + |B|.
De Morgan's Laws
Complement of a union: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ.
Complement of an intersection: (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.
Set Theoretic Difference
Definition
: A \ B = {x | x ∈ A and x ∉ B}.
Complement
: The complement of B with respect to A is A \ B.
Universal Set (U): The set containing all relevant elements for a specific topic.
Power Sets and Indexed Families of Sets
Power Set
: Contains all subsets of a set A.
Example: If A = {0, 1}, then P(A) = {∅, {0}, {1}, {0, 1}}.
Indexed Families of Sets
: Sets indexed by numbers (e.g., A_i for i in {1, 2, 3}).
Russell's Paradox
Discusses contradictions in set theory when considering sets that contain themselves.
Axiomatic set theory provides a rigorous framework to avoid such paradoxes.
Conclusion
Set theory provides a foundational structure for mathematics, yet it can lead to complex logical challenges.
Axiomatic approaches help clarify definitions and avoid paradoxes.
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