"A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind."
Introduces the concept of sets and elements, subsets, and set operations such as union, intersection, and complement.
Basics of sets: Introduction to sets and elements of sets, set notation, subset and superset, and universal set.
Operations on sets: Intersection, union, and complement of sets, Venn diagrams, and De Morgan's laws.
Binary relations: Definition of binary relations, equivalence relations, partial order relations, and Hasse diagrams.
Set families and partitions: Definition of set families and partition, equivalence classes and quotient set.
Cardinality: Cardinality of sets, countable and uncountable sets, and the cardinality of infinite sets.
Axiomatic set theory: Formal set theories, Zermelo-Fraenkel axioms and their relationship to the notion of infinity.
Applications of set theory: Applications in computer science, logic, and mathematics, set-theoretic topology, and set-theoretic models in physics and linguistics.
Set theory and logic: Basic concepts of logic, propositional logic and predicate logic, the connection between set theory and logic.
Combinatorics: Basic combinatorial concepts, permutations, combinations and binomial coefficients, and binomial theorem.
Measure theory: Introduction to measure theory and Lebesgue measure, measurable functions.
Topology: Basic concepts of topology, cover systems, compactness and connectedness.
Algebra of sets: Laws of algebra of sets, symmetrical difference and ordered pairs.
Order theory: Antisymmetric property, linear order and dense order.
Set: A collection of non-repeating elements.
Element: A single object within a set.
Empty set: A set containing no elements.
Universal set: A set containing all possible elements.
Subset: A set contained entirely within another set.
Complement: The elements not in a set.
Intersection: The set of elements common to two or more sets.
Union: The set of all elements in two or more sets.
Set Operations: Addition, subtraction, multiplication and division of sets.
Power Set: The set of all subsets of a set.
Cardinality: The number of elements in a set.
Finite and Infinite Sets: Sets with a limited number of elements and sets with an unlimited number of elements, respectively.
Singleton Set: A set containing only one element.
Ordered pairs: A set consisting of two elements in a certain order.
Sequences: An ordered list of elements, either finite or infinite.
Cartesian Product: The set of all possible ordered pairs made up of elements from two sets.
Equivalence Relations: A relation between elements of a set that is reflexive, symmetric and transitive.
Mapping and Functions: The assignment of each element of one set to a unique element of another set.
Well-Ordering Principle: Every non-empty set of positive integers contains a least element.
Axiom of Choice: States that a set can be constructed from an infinite number of non-empty sets.
"a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets."
"The set with no element is the empty set."
"A set with a single element is a singleton."
"A set may have a finite number of elements."
"A set may be an infinite set."
"Two sets are equal if they have precisely the same elements."
"Sets are ubiquitous in modern mathematics."
"Set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century."
"Set theory has been the standard way to provide rigorous foundations for all branches of mathematics."
"A set contains elements or members."
"A set can include even other sets."
"A set contains elements or members, which can be mathematical objects of any kind."
"Elements can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables."
"A set contains elements or members, which can be... other geometrical shapes."
"The set with no element is the empty set."
"A set with a single element is a singleton."
"A set may have a finite number of elements or be an infinite set."
"Set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics."
"Set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century."