Set Theory

• A set is a collection of distinct objects, called elements or members.
• A set relation is a mathematical concept that describes the relationship between elements of two sets.
• The union of two sets A and B, denoted by A ∪ B, is the set that contains all elements that are in A or in B, or in both.
• A power set is a set that contains all possible subsets of a given set.
• The cardinality of a set is a measure of the "size" or number of elements in the set.
• Two sets A and B are said to be equivalent if there exists a bijection (one-to-one and onto function) between them.
• An infinite set is a set that has an unlimited number of elements.
• Set theory is a branch of mathematical logic that studies sets, which are collections of objects.
• Axiomatic set theory is a foundational branch of mathematics that studies sets and their properties based on a set of axioms.
• Sets are usually denoted by capital letters, and elements are denoted by lowercase letters.
• The order of elements in a set does not matter, and each element appears only once in a set.
• The empty set, denoted by ∅ or {}, is a set with no elements.
• If two sets have exactly the same elements, they are considered equal.
• A set can be described by listing its elements inside curly braces, separated by commas.
• Two sets have the same cardinality if there exists a one-to-one correspondence (bijection) between their elements.
• A set is said to be finite if its cardinality is a natural number (including zero).
• A set is said to be infinite if its cardinality is not a natural number.
• The cardinality of the empty set is zero.
• The cardinality of a set with one element is one.
• The power set of a set with n elements has 2^n elements.
• The empty set and the set itself are always included in the power set.
• The power set of an empty set is a set containing only the empty set.
• The power set of a set with one element contains two elements: the empty set and the set itself.
• The power set of a set with two elements contains four elements: the empty set, the set itself, and the two individual elements.
• In set theory, a relation between two sets A and B is a subset of the Cartesian product A x B.
• A relation R from set A to set B is denoted as R ⊆ A x B.
• A relation can be represented as a set of ordered pairs (a, b), where a is an element of A and b is an element of B.
• A relation can be empty, meaning there are no ordered pairs in the relation.
• A relation can be reflexive if every element in set A is related to itself.
• An equivalence relation on a set S is a relation that is reflexive, symmetric, and transitive.
• Equivalence classes partition a set into subsets such that elements in the same subset are equivalent to each other.
• The equivalence class of an element a in a set S, denoted [a], is the set of all elements in S that are equivalent to a.
• Equivalence classes are disjoint, meaning that they do not have any elements in common.
• The set of all equivalence classes of a set S under an equivalence relation is called the quotient set or the set of equivalence classes.
• Set theory has applications in various fields, including computer science, linguistics, and philosophy.
• In computer science, set theory is used in database management systems to organize and manipulate data.
• In linguistics, set theory is used to study the structure of language and analyze sentence patterns.
• In philosophy, set theory is used to analyze concepts, define relationships between objects, and study the foundations of mathematics.
• Set theory is also used in probability theory and statistics to model and analyze random events and outcomes.
• The cardinality of an infinite set is denoted by the symbol ℵ₀ (aleph-null).