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One-to-one Correspondence
In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a injective function, one-to-one (injective) and surjective function, onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal ...
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Bijection
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a bijection, bijective function, one-to-one correspondence, or invertible function, is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The ...
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Diffeomorphism
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a diffeomorphism is an isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... of smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...s. It is an invertible function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a ...
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Linear Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the term linear function refers to two distinct but related notions: * In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ... and related areas, a linear function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... whose graph Graph may refer to: Mathematics *Graph (discrete mathematics), a stru ...
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Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was used as its argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab .... That is, for being identity, the equality holds for all . Definition Formally, if is a set, the identity function on is defined to be that function with domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ... and codomain In mathematics Mathematics (fr ...
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Cricket
Cricket is a bat-and-ball gameBat-and-ball may refer to: *Bat-and-ball games Bat-and-ball games (or safe haven games) are field games played by two opposing teams, in which the action starts when the defending team throws a ball at a dedicated player of the attacking team, wh ... played between two teams of eleven players each on a field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ... at the centre of which is a pitch Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octaves ... with a wicket In cricket Cricket is a Bat-and-ball games, bat-and-ball game played between tw ...
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Baseball
Baseball is a bat-and-ball gameBat-and-ball may refer to: *Bat-and-ball games Bat-and-ball games (or safe haven games) are field games played by two opposing teams, in which the action starts when the defending team throws a ball at a dedicated player of the attacking team, wh ... played between two opposing teams who take turns battingBatting may refer to: *Batting (baseball), the act of attempting to hit a ball thrown by the pitcher with a baseball bat, in order to score runs *Batting (cricket), the act of defending one's wicket with the cricket bat while attempting to score run ... and fielding. The game proceeds when a player on the fielding team, called the pitcher In baseball, the pitcher is the player who pitches the Baseball (ball), baseball from the pitcher's mound toward the catcher to begin each play, with the goal of out (baseball), retiring a batter (baseball), batter, who attempts to either make ..., throws a ball which a player on the batting t ...
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Batting Order (baseball)
In baseball Baseball is a bat-and-ball gameBat-and-ball may refer to: *Bat-and-ball games Bat-and-ball games (or safe haven games) are field games played by two opposing teams, in which the action starts when the defending team throws a ball at ..., the batting order or batting lineup is the sequence in which the members of the offense take their turns in battingBatting may refer to: *Batting (baseball), the act of attempting to hit a ball thrown by the pitcher with a baseball bat, in order to score runs *Batting (cricket), the act of defending one's wicket with the cricket bat while attempting to score run ... against the pitcher In baseball, the pitcher is the player who pitches the Baseball (ball), baseball from the pitcher's mound toward the catcher to begin each play, with the goal of out (baseball), retiring a batter (baseball), batter, who attempts to either make .... The batting order is the main component of a team's offensive Offensive may refer to ...
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Injective Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an injective function (also known as injection, or one-to-one function) is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... that maps distinct elements to distinct elements; that is, implies . In other words, every element of the function's codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... is ...
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One-to-one Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an injective function (also known as injection, or one-to-one function) is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... that maps distinct elements to distinct elements; that is, implies . In other words, every element of the function's codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... is ...
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Surjective Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a surjective function (also known as surjection, or onto function) is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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Onto
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a surjective function (also known as surjection, or onto function) is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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Domain Of A Function
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the domain of a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is the function. More precisely, given a function f\colon X\to Y, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both subsets of \R, the function can be graphed in the Cartesian coordinate system A Cartesian ...
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