Set (2)

An Euler diagram is a graphical representation of a collection of sets, each depicted as a planar region enclosed by a loop and containing its elements. If set A is a subset of set B, the region representing A is completely contained within the region representing B. If two sets have no elements in common, their corresponding regions do not overlap. In contrast, a Venn diagram is a graphical representation of n sets in which the n loops divide the plane into 2n zones. These zones represent all possible combinations of selecting some or none of the n sets. For example, if the sets are A, B, and C, there should be a zone for the elements that belong to A and C but not B, even if such elements do not exist.

There are certain sets of mathematical importance that are referred to so frequently that they have acquired special names and notations to identify them. These include the sets of natural numbers (N or Ν), integers (Z or Ζ), rational numbers (Q or Κ), real numbers (R or Ρ), and complex numbers (C or Ξ). These sets are usually represented in mathematical texts using bold or blackboard bold typeface. The sets of natural and integer numbers are infinite, while the other sets contain both finite and infinite numbers. In addition, each of these sets is a subset of the set listed below it. Sets of positive or negative numbers are sometimes denoted by a superscript plus or minus sign, respectively. For example, Q^+ represents the set of positive rational numbers.

A function is a rule that assigns a unique output to each input element in a set A, resulting in a set B. There are three types of functions: injective (or one-to-one), surjective (or onto), and bijective (or a one-to-one correspondence). An injective function is one in which no two different elements of set A are mapped to the same element in set B. A surjective function is one in which every element in set B is paired with at least one element in set A. A bijective function is both injective and surjective, meaning that each element in set A is paired with a unique element in set B and vice versa, resulting in no unpaired elements. An injective function is also known as an injection, a surjective function is known as a surjection, and a bijective function is known as a bijection or one-to-one correspondence.

A set is a collection of elements, also known as members. Two sets are equal if they have the same elements, meaning that every element of one set is also an element of the other set and vice versa. This property is known as extensionality. However, certain paradoxes can arise when constructing sets without any restrictions, such as Russell's paradox, which states that the set of all sets that do not contain themselves cannot exist, and Cantor's paradox, which shows that the set of all sets cannot exist. To resolve these paradoxes, set theory has been defined by axioms, which provide a framework for deducing the truth or falsity of particular mathematical statements about sets using first-order logic.

In mathematics, sets are commonly denoted by capital letters in italic, such as A, B, and C. Set-builder notation and roster notation are two ways of defining sets. Set-builder notation specifies a set by describing a condition that the elements must satisfy, while roster notation lists all the elements of the set between curly brackets. There are also several types of definitions in philosophy, including intensional definitions, which use a rule to determine membership, extensional definitions, which list all the elements of a set, and ostensive definitions, which give examples of elements. If every element of set A is also in set B, then A is a subset of B and is written as A⊆B. The empty set, denoted by ∅ or {} or ϕ, is a unique set that has no members. A singleton set is a set with only one element.

A function is a rule that assigns an element of set B to each element of set A. A function is injective if it maps any two different elements of A to different elements of B, surjective if it maps every element of B to at least one element of A, and bijective if it is both injective and surjective. The cardinality of a set, represented by |S|, is the number of members in the set. Two sets have the same cardinality if there exists a one-to-one correspondence between them. The cardinality of the empty set is zero, and the cardinality of infinite sets is infinite. Countable sets have a cardinality less than or equal to that of the natural

A set is a collection of elements, which can be represented in roster notation by listing the elements between curly brackets and separating them with commas. An infinite set is a set with an endless list of elements and is represented with an ellipsis at the end or both ends of the list. A set can also be defined by a rule that determines its elements, called a semantic description. Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. A function is a rule that assigns an output element in set B to each input element in set A. The cardinality of a set is the number of elements it contains. The power set of a set is the set of all subsets of that set. A partition of a set is a set of nonempty subsets of the set, such that every element in the set is contained in exactly one of the subsets and the subsets are pairwise disjoint.

The complement of a set A is a set of all elements that are not in A. It is denoted as Ac or A'. For example, if the universal set (a set containing all elements being discussed) is the set of integers, then the complement of the set of even integers would be the set of odd integers. The union of two sets A and B is the set of all elements that belong to either A or B or both. The intersection of A and B is the set of all elements that belong to both A and B. If the intersection of A and B is the empty set, then A and B are said to be disjoint. The set difference of A and B, denoted as A \ B or A - B, is the set of all elements that belong to A but not B. It is also called the relative complement of B in A when B is a subset of A. The symmetric difference of A and B, denoted as A Δ B, is the set of all elements that belong to A or B but not both. The cartesian product of A and B, denoted as A × B, is the set of all ordered pairs (a,b) such that a is an element of A and b is an element of B. The cardinality of A × B is the product of the cardinalities of A and B. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.

Sets are an integral part of modern mathematics, appearing in various structures such as groups, rings, and fields. One common application of set theory is in the construction of relations. For example, consider a set S that represents the shapes in the game rock-paper-scissors. The relation "beats" from S to S would be the set of all pairs of shapes that can beat each other in the game, such as (scissors, paper). This relation is a subset of S x S, the Cartesian product of S with itself. Another example is the set of all pairs (x, x^2) where x is a real number. This relation, written as a function F(x) = x^2, is a subset of the set of real numbers R x R.

The inclusion-exclusion principle is a method for counting the elements in the union of two finite sets by considering the sizes of the sets and their intersection. It can be expressed as |A U B| = |A| + |B| - |A intersection B|. A more general form of the principle gives the cardinality of any finite union of finite sets: |A1 U A2 U A3 ... Un| = (|A1| + |A2| + |A3| + ... |An|) - (|A1 intersection A2| + |A1 intersection A3| + ... |An-1 intersection An|) + ... + (-1)^(n-1) * (|A1 intersection A2 intersection A3 ... intersection An|).