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Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It serves as the foundation for modern mathematics, providing a rigorous framework for defining mathematical structures and reasoning about infinity.

Set Theory Topics

Set

Set

A well-defined collection of distinct objects, considered as an object in its own right. The fundamental concept of set theory.

A well-defined collection of distinct objects, considered as an object in its own right. The fundamental concept of set theory. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Set.

Basic Set Operations

The fundamental operations on sets are: union (A ∪ B), intersection (A ∩ B), difference (A \ B), and complement (Aᶜ). De Morgan's laws relate these: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ. The power set P(A) is the set of all subsets of A; if |A| = n then |P(A)| = 2ⁿ.

\[ |A \cup B| = |A| + |B| - |A \cap B| \]

Relations and Functions

A relation on a set A is a subset of A × A. An equivalence relation is reflexive, symmetric, and transitive; it partitions A into equivalence classes. A function f: A → B assigns exactly one element of B to each element of A. Functions can be injective (one-to-one), surjective (onto), or bijective (both).

Cardinality

Two sets have the same cardinality if there exists a bijection between them. Countably infinite sets (like ℕ and ℤ) have cardinality ℵ₀. Cantor's diagonal argument proves that ℝ is uncountable — its cardinality (the continuum) is strictly greater than ℵ₀. This shows there are different "sizes" of infinity.

\[ |\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| = \aleph_0 < |\mathbb{R}| \]

Zermelo-Fraenkel Axioms

Modern set theory is built on the Zermelo-Fraenkel axioms (ZF), often supplemented by the Axiom of Choice (ZFC). These axioms provide a rigorous foundation for all of mathematics, defining what sets exist and how they can be constructed. The Axiom of Choice, while controversial, is equivalent to many important theorems including Zorn's Lemma and the Well-Ordering Theorem.

Subset

Subset

A set A is a subset of B if every element of A is also an element of B. Denoted as A ⊆ B.

A set A is a subset of B if every element of A is also an element of B. Denoted as A ⊆ B. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Subset.

Union

Union

The union of two sets contains all elements that are in either set. Denoted as A ∪ B.

The union of two sets contains all elements that are in either set. Denoted as A ∪ B. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Union.

Intersection

Intersection

The intersection of two sets contains only elements that are in both sets. Denoted as A ∩ B.

The intersection of two sets contains only elements that are in both sets. Denoted as A ∩ B. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Intersection.

Complement

Complement

The complement of a set contains all elements not in the set, relative to a universal set. Denoted as A' or A^c.

The complement of a set contains all elements not in the set, relative to a universal set. Denoted as A' or A^c. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Complement.

Power Set

Power Set

The set of all subsets of a given set. If A has n elements, its power set has 2^n elements. Denoted as P(A).

The set of all subsets of a given set. If A has n elements, its power set has 2^n elements. Denoted as P(A). This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Power Set.

Cartesian Product

Cartesian Product

The set of all ordered pairs (a,b) where a ∈ A and b ∈ B. Forms the basis for relations and functions. Denoted as A × B.

The set of all ordered pairs (a,b) where a ∈ A and b ∈ B. Forms the basis for relations and functions. Denoted as A × B. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Cartesian Product.

Relation

Relation

A subset of the Cartesian product of two sets. Relations describe connections between elements of different sets.

A subset of the Cartesian product of two sets. Relations describe connections between elements of different sets. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Relation.

Function

Function

A special relation where each element of the domain maps to exactly one element of the codomain. Central to all mathematics.

A special relation where each element of the domain maps to exactly one element of the codomain. Central to all mathematics. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Function.

Cardinality

Cardinality

A measure of the number of elements in a set. Leads to the study of finite, countable, and uncountable sets.

A measure of the number of elements in a set. Leads to the study of finite, countable, and uncountable sets. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Cardinality.

Countable Set

Countable Set

A set whose elements can be put into one-to-one correspondence with natural numbers. Includes finite sets and sets like integers and rationals.

A set whose elements can be put into one-to-one correspondence with natural numbers. Includes finite sets and sets like integers and rationals. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

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Uncountable Set

Uncountable Set

A set that cannot be put into one-to-one correspondence with natural numbers. The real numbers form the canonical example.

A set that cannot be put into one-to-one correspondence with natural numbers. The real numbers form the canonical example. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Uncountable Set.