"In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y."
Relates the concepts of relations and functions to sets, including equivalence relations, partial orderings, and well-orderings.
Sets: Sets are collections of objects, and they form the basis of set theory. Understanding the properties of sets and how they are manipulated is important when studying relations and functions.
Relations: Relations are sets of ordered pairs, where each pair relates two elements from two different sets. They are used to study properties of functions.
Functions: Functions are a type of relation that assigns each element of one set to a unique element in another set. They are used to describe mathematical relationships.
Cartesian products: The Cartesian product of two sets is a set of all ordered pairs where the first element is from the first set and the second element is from the second set. This concept is important when studying relations and functions.
Domains and ranges: The domain of a function is the set of all inputs, while the range is the set of all outputs. Understanding these concepts is important when studying functions.
Composing functions: Composing functions is a way to combine two functions to create a new function. This concept is important for understanding complex mathematical relationships.
Inverses: The inverse of a function is another function that undoes the original function. Understanding how to find and use inverse functions is important when studying functions.
Injective, surjective, and bijective functions: These are different types of functions that determine certain properties of the function. Injective functions are one-to-one, while surjective functions are onto. Bijective functions are both one-to-one and onto.
Equivalence relations: Equivalence relations are relations that have certain properties, such as reflexivity, symmetry, and transitivity. These relations are important for understanding the relationship between different elements in a set.
Partial orders: Partial orders are relations that have certain properties, such as reflexivity, antisymmetry, and transitivity. They are important for understanding how to order and compare different elements in a set.
Reflexive Relation: A relation is said to be reflexive if every element in the set is related to itself. For example, the relation “is equal to” is reflexive, as every element is equal to itself.
Symmetric Relation: A relation is said to be symmetric if for every pair of elements in the set (a, b), the relation between them is the same as the relation between (b, a). For example, the relation “is the same as” is symmetric, as if a is the same as b, then b is also the same as a.
Antisymmetric Relation: A relation is said to be antisymmetric if for every distinct pair of elements in the set (a, b), if a is related to b, then b is not related to a. For example, the relation “is less than” is antisymmetric, as if a is less than b, then b is not less than a.
Transitive Relation: A relation is said to be transitive if for every triplet of elements in the set (a, b, c), if a is related to b and b is related to c, then a is also related to c. For example, the relation “is smaller than” is transitive, as if a is smaller than b and b is smaller than c, then a is also smaller than c.
Equivalence Relation: An equivalence relation is a relation that is reflexive, symmetric, and transitive. For example, the relation “is congruent to” is an equivalence relation, as every number is congruent to itself, if a is congruent to b then b is congruent to a, and if a is congruent to b and b is congruent to c, then a is congruent to c.
Partial Order Relation: A partial order relation is a relation that is reflexive, antisymmetric, and transitive. For example, the relation “is less than or equal to” is a partial order relation.
Total Order Relation: A total order relation is a partial order relation that is also a total relation, meaning that for every distinct pair of elements in the set (a, b), either a is related to b or b is related to a. For example, the relation “is less than or equal to” is a total order relation.
One-to-one Function: A one-to-one function is a function in which every element in the domain is assigned to a unique element in the range. For example, the function f(x) = 2x is one-to-one, as every element in the domain is assigned to a unique element in the range.
Onto Function: An onto function is a function in which every element in the range is assigned to by at least one element in the domain. For example, the function f(x) = x^2 is onto, as every element in the range is assigned to by at least one element in the domain.
Bijective Function: A bijective function is a function that is both one-to-one and onto. For example, the function f(x) = x is bijective, as every element in the domain is assigned to a unique element in the range and every element in the range is assigned to by at least one element in the domain.
"The set X is called the domain of the function and the set Y is called the codomain of the function."
"Functions were originally the idealization of how a varying quantity depends on another quantity."
"For example, the position of a planet is a function of time."
"Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century."
"Until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity)."
"The concept of a function was formalized at the end of the 19th century in terms of set theory."
"A function is most often denoted by letters such as f, g and h."
"The value of a function f at an element x of its domain is denoted by f(x)."
"The numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value."
"For example, the value of f at x = 4 is denoted by f(4)."
"When the function is not named and is represented by an expression E, the value of the function at, say, x = 4 may be denoted by E|x=4."
"For example, the value at 4 of the function that maps x to x^2 may be denoted by x^2|x=4."
"A function is uniquely represented by the set of all pairs (x, f(x)), called the graph of the function."
"The graph of the function [pairs (x, f(x))] is a popular means of illustrating the function."
"Functions are widely used in science, engineering, and in most fields of mathematics."
"It has been said that functions are 'the central objects of investigation' in most fields of mathematics."
"The concept of a function was formalized...greatly enlarging the domains of application of the concept."
"Functions are widely used in science, engineering, and in most fields of mathematics."
"Functions are 'the central objects of investigation' in most fields of mathematics."