Injective functionIn mathematics, an injective function (also known as injection, or one-to-one function) is a functionf that maps distinct elements of its domain to distinct elements of its codomain; that is, x1 ≠ x2 implies f(x1) ≠ f(x2) (equivalently by contraposition, f(x1) = f(x2) implies x1 = x2). In other words, every element of the function's codomain is the image of at most one element of its domain.
CodomainIn mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in the notation f : X → Y. The term range is sometimes ambiguously used to refer to either the codomain or the image of a function. A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph.
Domain of a functionIn mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom(f){\displaystyle \operatorname {dom} (f)} or domf{\displaystyle \operatorname {dom} f}, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".
Image (mathematics)Set of the values of a functionFor the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}.
Function (mathematics)In mathematics, a function from a setX to a set Y assigns to each element of X exactly one element of Y. 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.
BijectionIn mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given a function f:A→B{\displaystyle f:A\to B}, the image of an elementa∈A{\displaystyle a\in A} is the element f(a)∈B{\displaystyle f(a)\in B} in the codomain.
Restriction (mathematics)In mathematics, the restriction of a functionf{\displaystyle f} is a new function, denoted f|A{\displaystyle f\vert _{A}} or f↾A,{\displaystyle f{\upharpoonright _{A}},} obtained by choosing a smaller domainA{\displaystyle A} for the original function f.{\displaystyle f.} The function f{\displaystyle f} is then said to extendf|A.{\displaystyle f\vert _{A}.}Formal definitionLet f:E→F{\displaystyle f:E\to F} be a function from a setE{\displaystyle E} to a set F.{\displaystyle F.} If a set A{\displaystyle A} is a subset of E,{\displaystyle E,} then the restriction of f{\displaystyle f}to A{\displaystyle A} is the function f|A:A→F{\displaystyle {f|}_{A}:A\to F} given by f|A(x)=f(x){\displaystyle {f|}_{A}(x)=f(x)} for x∈A.{\displaystyle x\in A.} Informally, the restriction of f{\displaystyle f} to A{\displaystyle A} is the same function as f,{\displaystyle f,} but is only defined on A{\displaystyle A}.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.
Uniqueness quantificationIn mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" or "∃=1". It is defined to mean there exists an object with the given property, and all objects with this property are equal.
MathematicianA mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. HistoryOne of the earliest known mathematicians was Thales of Miletus (c. 624 – c. 546 BC); He has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.
FranceCountry primarily in Western EuropeFrance, officially the French Republic, is a country primarily located in Western Europe. Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the North Atlantic, the French West Indies, and many islands in Oceania and the Indian Ocean.
Nicolas BourbakiPseudonym of a group of mathematiciansNicolas Bourbaki (French:) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics.ontoRelated word or term.surjectiveRelated word or term.surjectionRelated word or term.*As an Amazon Associate I earn from qualifying purchases.