A set is a collection of distinct objects, called elements or members of the set.
The elements of a set are not necessarily ordered and they can be of any type (numbers, letters, other sets, etc.).
Sets are usually denoted by capital letters, such as A, B, or S, and the elements of a set are enclosed in curly brackets {}. For example, {1, 2, 3} is a set with three elements: 1, 2, and 3.
A set can be empty, called the empty set or null set, denoted by {} or the symbol ∅.
Sets can be compared using set-theoretic operations such as union, intersection, and complement.
Union of two sets A and B is the set of all elements that are in A or in B (or in both). It is denoted by A ∪ B.
Intersection of two sets A and B is the set of all elements that are in both A and B. It is denoted by A ∩ B.
Complement of a set A with respect to a set B is the set of all elements in B that are not in A. It is denoted by A' or A^c.
Subsets are sets that contain only elements from another set. A set A is a subset of B, if every element of A is also an element of B. It is denoted by A ⊆ B.
A proper subset is a subset that contains strictly less elements than the set it is a subset of. A is a proper subset of B, if A⊆B and A≠B
The power set of a set A is the set of all subsets of A, including the empty set and A itself. It is denoted by P(A)
Sets can be infinite or finite, cardinality is the number of elements in a set
Set theory forms the foundation of modern mathematics, and many branches of mathematics like algebra, topology, and analysis use set-theoretic concepts and notation.
Zermelo-Fraenkel axioms(ZF) is the standard system of axioms for set theory, which provides a foundation for the study of sets and their properties.
It's important to note that set theory is a vast and complex field with many different branches and subtopics. It is a foundation of modern mathematics and is used in many branches of mathematics, as well as computer science and logic
Sets have many properties that are important to understand when working with them in mathematics and other fields. Some important properties of sets include:
Extensionality: Two sets are equal if and only if they have the same elements.
Pairwise disjointness: Two sets are disjoint if they have no elements in common.
Idempotence: The union or intersection of a set with itself is equal to the set itself.
Commutativity: The order of the sets being unioned or intersected does not matter, i.e, A ∪ B = B ∪ A, A ∩ B = B ∩ A
Associativity: The order in which the union or intersection of multiple sets is performed does not matter. i.e. A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C
Distribution: The intersection of a set with the union of multiple sets is the union of the intersections of the set with each of the multiple sets. i.e. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan's laws: The complement of the union of two sets is equal to the intersection of their complements, and the complement of the intersection of two sets is equal to the union of their complements.
Empty set: The empty set is a subset of every set and the intersection of any set with the empty set is the empty set.
Universal set: A universal set is a set that contains all the elements of other sets under consideration.
Power set: The power set of a set A is the set of all subsets of A, including the empty set and A itself.
