Set (1)

The concept of a set is a fundamental building block in mathematics, representing a collection of distinct objects or elements. These elements can be any mathematical objects - numbers, symbols, shapes, or even other sets. A set with no elements is called an empty set, while a set with a single element is called a singleton. Sets can be finite or infinite in size, and are considered equal if they have the same elements. In modern mathematics, set theory, particularly Zermelo-Fraenkel set theory, plays a crucial role in providing a solid foundation for all areas of math.

The concept of a set first appeared in mathematics in the late 19th century. The German word for set, "Menge," was introduced by Bernard Bolzano in his work "Paradoxes of the Infinite." Georg Cantor, one of the founders of set theory, defined a set as "a gathering together into a whole of definite, distinct objects of our perception or our thought - which are called elements of the set." Bertrand Russell referred to a set as a "class," saying that when mathematicians deal with a set, they often consider it "defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class."

One of the most important properties of a set is that it can contain elements, also known as members. Two sets are equal if they have the same elements. This property, called extensionality, states that for sets A and B to be equal, every element in A must also be an element in B, and every element in B must also be an element in A. While the concept of a set has proven to be extremely useful in mathematics, there are certain paradoxes that arise when no restrictions are placed on how sets can be constructed. These paradoxes, such as Russell's paradox and Cantor's paradox, highlight the importance of defining sets in a precise and well-defined manner.

To address the paradoxes that arose in the original formulation of naive set theory, the properties of sets have been defined using axioms. Axiomatic set theory views the concept of a set as a primitive idea and uses a set of axioms to establish a basic framework for determining the truth or falsity of statements about sets using first-order logic. However, according to Gödel's incompleteness theorems, it is not possible to use first-order logic to prove that any particular axiomatic set theory is free from paradoxes.

In mathematics, sets are commonly represented by capital letters in italic font, such as A, B, C. Sets may also be referred to as collections or families, particularly when their elements are themselves sets. To define a set, its elements are listed between curly brackets and separated by commas, a method known as roster notation. For example, the set {2, 4, 6} and {4, 6, 4, 2} represent the same set, as the ordering of the elements is not important in a set (unlike in a sequence, tuple, or permutation). In cases where a set has many elements, it can be abbreviated using an ellipsis, such as the set of the first thousand positive integers, which can be written as {1, 2, 3, ..., 1000}.

An infinite set is a set that contains an infinite number of elements. In roster notation, an infinite set is represented by listing a portion of its elements and then using an ellipsis to indicate that the list continues indefinitely. For example, the set of nonnegative integers is written as {0, 1, 2, 3, 4, ...}, and the set of all integers is written as {..., -3, -2, -1, 0, 1, 2, 3, ...}. Another way to define a set is to provide a rule that determines what elements belong to the set. For example, "Let A be the set whose members are the first four positive integers" and "Let B be the set of colors of the French flag." This type of definition is called a semantic description.

Set-builder notation is a way to define a set by specifying a condition that elements in the set must satisfy. For example, the set F can be defined as follows: F = {n | n is an integer and 0 ≤ n ≤ 19}. In this notation, the vertical bar "|" means "such that," and the description can be read as "F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive." Some authors use a colon ":" instead of the vertical bar. There are several types of definitions used in philosophy to classify different types of sets. Intensional definitions use a rule to determine membership, while extensional definitions describe a set by listing all of its elements. Ostensive definitions describe a set by providing examples of its elements.

In mathematics, if B is a set and x is an element of B, this is written as x ∈ B, which can be read as "x belongs to B" or "x is in B." The statement "y is not an element of B" is written as y ∉ B, which can be read as "y is not in B." For example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white, red}, and F = {n | n is an integer and 0 ≤ n ≤ 19}, we have 4 ∈ A and 12 ∈ F, but 20 ∉ F and green ∉ B. The empty set, also known as the null set, is a set that has no elements. It is denoted by ∅, { }, or ϕ. A singleton set is a set that contains exactly one element. It can be written as {x}, where x is the element. It is important to note that the set {x} and the element x are not the same thing.

If every element in set A is also an element of set B, then A is said to be a subset of B, written A ⊆ B, or B is a superset of A, written B ⊇ A. This relationship between sets is known as inclusion or containment. If A is a subset of B but is not equal to B, then A is called a proper subset of B, written A ⊊ B. Similarly, B is a proper superset of A, written B ⊋ A, if it contains A but is not equal to A. There are also two other pairs of operators, ⊂ and ⊃, that are used differently by different authors. Some authors use ⊂ and ⊃ to indicate any subset (not necessarily a proper subset), while others reserve them for indicating proper subsets. It is important to note that the empty set is a subset of every set and every set is a subset of itself.