Set
Definition and Basic Concepts
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are fundamental objects in mathematics and are used to define most mathematical structures. The concept of a set is one of the most basic in mathematics, and it serves as a foundation for various branches such as Set Theory, Algebra, and Topology.
A set is typically denoted by a capital letter, and the objects within the set are called its elements or members. For example, the set of natural numbers is denoted by \( \mathbb{N} \), and it includes elements like 1, 2, 3, and so on. The notation \( a \in A \) denotes that \( a \) is an element of the set \( A \).
Sets can be described in two main ways: by listing their elements, known as the roster method, or by specifying a property that its members satisfy, known as the set-builder notation. For instance, the set of all even numbers can be written as \( \{2, 4, 6, 8, \ldots\} \) or \( \{x \mid x \text{ is an even number}\} \).
Types of Sets
Finite and Infinite Sets
A set is called finite if it contains a limited number of elements. For example, the set \( \{1, 2, 3, 4, 5\} \) is finite because it contains exactly five elements. In contrast, a set is infinite if it has an unlimited number of elements. The set of all natural numbers \( \mathbb{N} \) is an example of an infinite set.
Subsets
A set \( A \) is a subset of a set \( B \) if every element of \( A \) is also an element of \( B \). This is denoted as \( A \subseteq B \). If \( A \) is a subset of \( B \) but not equal to \( B \), then \( A \) is a proper subset of \( B \), denoted as \( A \subset B \).
Power Sets
The power set of a set \( A \), denoted as \( \mathcal{P}(A) \), is the set of all possible subsets of \( A \). For a set with \( n \) elements, the power set contains \( 2^n \) elements. For example, if \( A = \{1, 2\} \), then \( \mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} \).
Universal Sets and Complements
A universal set is a set that contains all the objects under consideration, typically denoted by \( U \). The complement of a set \( A \), denoted by \( A^c \) or \( \overline{A} \), is the set of all elements in the universal set that are not in \( A \).
Operations on Sets
Union
The union of two sets \( A \) and \( B \), denoted \( A \cup B \), is the set of elements that are in \( A \), in \( B \), or in both. Formally, \( A \cup B = \{x \mid x \in A \text{ or } x \in B\} \).
Intersection
The intersection of two sets \( A \) and \( B \), denoted \( A \cap B \), is the set of elements that are in both \( A \) and \( B \). Formally, \( A \cap B = \{x \mid x \in A \text{ and } x \in B\} \).
Difference
The difference of two sets \( A \) and \( B \), denoted \( A - B \) or \( A \setminus B \), is the set of elements that are in \( A \) but not in \( B \). Formally, \( A - B = \{x \mid x \in A \text{ and } x \notin B\} \).
Symmetric Difference
The symmetric difference of two sets \( A \) and \( B \), denoted \( A \Delta B \), is the set of elements that are in either of the sets but not in their intersection. Formally, \( A \Delta B = (A - B) \cup (B - A) \).
Special Sets
Empty Set
The empty set, denoted by \( \emptyset \) or \(\{\}\), is the unique set having no elements. It is a subset of every set and is important in the foundation of set theory.
Singleton Set
A singleton set is a set with exactly one element. For example, the set \(\{a\}\) is a singleton set.
Cartesian Product
The Cartesian product of two sets \( A \) and \( B \), denoted \( A \times B \), is the set of all ordered pairs \((a, b)\) where \( a \in A \) and \( b \in B \). Formally, \( A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\} \).
Set Theory
Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It was developed in the late 19th century by Georg Cantor, who introduced the concept of infinite sets and cardinality. Set theory forms the basis for several other fields of mathematics and is used to define numbers, functions, and other mathematical objects.
Axiomatic Set Theory
Axiomatic set theory is a formal system that provides a rigorous foundation for the study of sets. The most commonly used system is Zermelo-Fraenkel set theory (ZF), often with the Axiom of Choice (ZFC). These axioms avoid certain paradoxes, such as Russell's Paradox, by carefully defining the properties and operations of sets.
Cardinality
The cardinality of a set is a measure of the "number of elements" in the set. For finite sets, the cardinality is simply the number of elements. For infinite sets, cardinality is determined by the concept of bijection, which allows comparison of the sizes of infinite sets. The cardinality of the set of natural numbers is denoted by \( \aleph_0 \), and larger infinities are represented by larger cardinal numbers.
Applications of Sets
Sets are used extensively in various fields of mathematics and science. In Probability Theory, sets are used to define events and sample spaces. In Computer Science, sets are used in data structures and algorithms. In Linguistics, sets are used to model languages and grammars.