A proposition is a central concept in the philosophy of language , semantics , logic , and related fields, often characterized as the primary bearer of truth or falsity . Propositions are also often characterized as the type of object that declarative sentences denote . For instance the sentence "The sky is blue" denotes the proposition that the sky is blue. However, crucially, propositions are not themselves linguistic expressions . For instance, the English sentence "Snow is white" denotes the same proposition as the German sentence "Schnee ist weiß" even though the two sentences are not the same. Similarly, propositions can also be characterized as the objects of belief and other propositional attitudes . For instance if someone believes that the sky is blue, the object of their belief is the proposition that the sky is blue.
92-400: Formally, propositions are often modeled as functions which map a possible world to a truth value . For instance, the proposition that the sky is blue can be modeled as a function which would return the truth value T {\displaystyle T} if given the actual world as input, but would return F {\displaystyle F} if given some alternate world where
184-538: A ∈ A . {\displaystyle a\in A.} It is denoted by f [ A ] , {\displaystyle f[A],} or by f ( A ) , {\displaystyle f(A),} when there is no risk of confusion. Using set-builder notation , this definition can be written as f [ A ] = { f ( a ) : a ∈ A } . {\displaystyle f[A]=\{f(a):a\in A\}.} This induces
276-511: A dual pair to show the underlying duality . This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation , the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application . In category theory and homological algebra , networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize
368-432: A map or a mapping , but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds ). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H ). Some authors reserve
460-525: A possible world and return a truth value. For example, the proposition that the sky is blue could be represented as a function f {\displaystyle f} such that f ( w ) = T {\displaystyle f(w)=T} for every world w , {\displaystyle w,} if any, where the sky is blue, and f ( v ) = F {\displaystyle f(v)=F} for every world v , {\displaystyle v,} if any, where it
552-460: A predicate of a subject , optionally with the help of a copula . An Aristotelian proposition may take the form of "All men are mortal" or "Socrates is a man." In the first example, the subject is "men", predicate is "mortal" and copula is "are", while in the second example, the subject is "Socrates", the predicate is "a man" and copula is "is". Often, propositions are related to closed formulae (or logical sentence) to distinguish them from what
644-405: A roman type is customarily used instead, such as " sin " for the sine function , in contrast to italic font for single-letter symbols. The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let f ( x ) {\displaystyle f(x)} be a function". This is an abuse of notation that is useful for
736-421: A singleton set , denoted by f − 1 [ { y } ] {\displaystyle f^{-1}[\{y\}]} or by f − 1 [ y ] , {\displaystyle f^{-1}[y],} is also called the fiber or fiber over y {\displaystyle y} or the level set of y . {\displaystyle y.} The set of all
828-533: A function f [ ⋅ ] : P ( X ) → P ( Y ) , {\displaystyle f[\,\cdot \,]:{\mathcal {P}}(X)\to {\mathcal {P}}(Y),} where P ( S ) {\displaystyle {\mathcal {P}}(S)} denotes the power set of a set S ; {\displaystyle S;} that is the set of all subsets of S . {\displaystyle S.} See § Notation below for more. The image of
920-429: A function defined by an integral with variable upper bound: x ↦ ∫ a x f ( u ) d u {\textstyle x\mapsto \int _{a}^{x}f(u)\,du} . There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis , linear forms and the vectors they act upon are denoted using
1012-401: A function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details. A function f from a set X to a set Y is an assignment of one element of Y to each element of X . The set X is called the domain of the function and the set Y is called the codomain of
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#17327660939941104-523: A function is commonly written as f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as f ( w , x , y ) {\displaystyle f(w,x,y)} , f ( w , x , y , z ) {\displaystyle f(w,x,y,z)} . A function may also be called
1196-513: A function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is a real function , the determination of the domain of the function x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} requires knowing
1288-421: A function is the image of its entire domain , also known as the range of the function. This last usage should be avoided because the word "range" is also commonly used to mean the codomain of f . {\displaystyle f.} If R {\displaystyle R} is an arbitrary binary relation on X × Y , {\displaystyle X\times Y,} then
1380-405: A function is then called a partial function . The range or image of a function is the set of the images of all elements in the domain. A function f on a set S means a function from the domain S , without specifying a codomain. However, some authors use it as shorthand for saying that the function is f : S → S . The above definition of a function is essentially that of
1472-781: A mental state of believing that it is raining, her mental content is the proposition 'it is raining.' Furthermore, since such mental states are about something (namely, propositions), they are said to be intentional mental states. Explaining the relation of propositions to the mind is especially difficult for non-mentalist views of propositions, such as those of the logical positivists and Russell described above, and Gottlob Frege 's view that propositions are Platonist entities, that is, existing in an abstract, non-physical realm. So some recent views of propositions have taken them to be mental. Although propositions cannot be particular thoughts since those are not shareable, they could be types of cognitive events or properties of thoughts (which could be
1564-417: A multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain. where the domain U has the form If all the X i {\displaystyle X_{i}} are equal to the set R {\displaystyle \mathbb {R} } of the real numbers or to the set C {\displaystyle \mathbb {C} } of
1656-406: A new car ", or "I wonder whether it will snow " (or, whether it is the case that "it will snow"). Desire, belief, doubt, and so on, are thus called propositional attitudes when they take this sort of content. Bertrand Russell held that propositions were structured entities with objects and properties as constituents. One important difference between Ludwig Wittgenstein 's view (according to which
1748-417: A proposition (e.g. 'it is raining,' 'snow is white,' etc.). In English, propositions usually follow folk psychological attitudes by a "that clause" (e.g. "Jane believes that it is raining"). In philosophy of mind and psychology , mental states are often taken to primarily consist in propositional attitudes. The propositions are usually said to be the "mental content" of the attitude. For example, if Jane has
1840-416: A proposition are too vague to be useful. For them, it is just a misleading concept that should be removed from philosophy and semantics . W. V. Quine , who granted the existence of sets in mathematics, maintained that the indeterminacy of translation prevented any meaningful discussion of propositions, and that they should be discarded in favor of sentences. P. F. Strawson , on the other hand, advocated for
1932-537: A proposition is the set of possible worlds /states of affairs in which it is true) is that on the Russellian account, two propositions that are true in all the same states of affairs can still be differentiated. For instance, the proposition "two plus two equals four" is distinct on a Russellian account from the proposition "three plus three equals six". If propositions are sets of possible worlds, however, then all mathematical truths (and all other necessary truths) are
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#17327660939942024-450: A simpler formulation. Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced " maps to ". For example, x ↦ x + 1 {\displaystyle x\mapsto x+1} is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of R {\displaystyle \mathbb {R} }
2116-413: A structured view of propositions, one can distinguish between singular propositions (also Russellian propositions , named after Bertrand Russell ) which are about a particular individual, general propositions , which are not about any particular individual, and particularized propositions , which are about a particular individual but do not contain that individual as a constituent. Attempts to provide
2208-465: A value if there exists some x {\displaystyle x} in the function's domain such that f ( x ) = y . {\displaystyle f(x)=y.} Similarly, given a set S , {\displaystyle S,} f {\displaystyle f} is said to take a value in S {\displaystyle S} if there exists some x {\displaystyle x} in
2300-590: A workable definition of proposition include the following: Two meaningful declarative sentences express the same proposition, if and only if they mean the same thing. which defines proposition in terms of synonymity. For example, "Snow is white" (in English) and "Schnee ist weiß" (in German) are different sentences, but they say the same thing, so they express the same proposition. Another definition of proposition is: Two meaningful declarative sentence-tokens express
2392-423: Is a function in two variables, and we want to refer to a partially applied function X → Y {\displaystyle X\to Y} produced by fixing the second argument to the value t 0 without introducing a new function name. The map in question could be denoted x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} using
2484-400: Is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. Formally, a function of n variables is a function whose domain is a set of n -tuples. For example, multiplication of integers is a function of two variables, or bivariate function , whose domain is
2576-454: Is a member of X , {\displaystyle X,} then the image of x {\displaystyle x} under f , {\displaystyle f,} denoted f ( x ) , {\displaystyle f(x),} is the value of f {\displaystyle f} when applied to x . {\displaystyle x.} f ( x ) {\displaystyle f(x)}
2668-431: Is a philosopher” and “Plato is a philosopher” are different propositions. Similarly, “I am Spartacus” becomes “X is Spartacus”, where X is replaced with terms representing the individuals Spartacus and John Smith. In other words, the example problems can be averted if sentences are formulated with precision such that their terms have unambiguous meanings. A number of philosophers and linguists claim that all definitions of
2760-406: Is alternatively known as the output of f {\displaystyle f} for argument x . {\displaystyle x.} Given y , {\displaystyle y,} the function f {\displaystyle f} is said to take the value y {\displaystyle y} or take y {\displaystyle y} as
2852-484: Is called the Cartesian product of X and Y and denoted X × Y . {\displaystyle X\times Y.} Thus, the above definition may be formalized as follows. A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions: This definition may be rewritten more formally, without referring explicitly to
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2944-527: Is called the domain of R . {\displaystyle R.} Let f {\displaystyle f} be a function from X {\displaystyle X} to Y . {\displaystyle Y.} The preimage or inverse image of a set B ⊆ Y {\displaystyle B\subseteq Y} under f , {\displaystyle f,} denoted by f − 1 [ B ] , {\displaystyle f^{-1}[B],}
3036-414: Is expressed by an open formula . In this sense, propositions are "statements" that are truth-bearers . This conception of a proposition was supported by the philosophical school of logical positivism . Some philosophers argue that some (or all) kinds of speech or actions besides the declarative ones also have propositional content. For example, yes–no questions present propositions, being inquiries into
3128-438: Is implied. The domain and codomain can also be explicitly stated, for example: This defines a function sqr from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f : X × X → Y ; ( x , t ) ↦ f ( x , t ) {\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}
3220-432: Is in Y , or it is undefined. The set of the elements of X such that f ( x ) {\displaystyle f(x)} is defined and belongs to Y is called the domain of definition of the function. A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X , one often says that
3312-427: Is no risk of confusion, f − 1 [ B ] {\displaystyle f^{-1}[B]} can be denoted by f − 1 ( B ) , {\displaystyle f^{-1}(B),} and f − 1 {\displaystyle f^{-1}} can also be thought of as a function from the power set of Y {\displaystyle Y} to
3404-400: Is not. A proposition can be modeled equivalently with the inverse image of T {\displaystyle T} under the indicator function, which is sometimes called the characteristic set of the proposition. For instance, if w {\displaystyle w} and w ′ {\displaystyle w'} are the only worlds in which the sky is blue,
3496-454: Is often the letter f . Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in The argument between the parentheses may be a variable , often x , that represents an arbitrary element of the domain of the function, a specific element of the domain ( 3 in
3588-449: Is rarely used. Image and inverse image may also be defined for general binary relations , not just functions. The word "image" is used in three related ways. In these definitions, f : X → Y {\displaystyle f:X\to Y} is a function from the set X {\displaystyle X} to the set Y . {\displaystyle Y.} If x {\displaystyle x}
3680-440: Is the value of the function at x , or the image of x under the function. A function f , its domain X , and its codomain Y are often specified by the notation f : X → Y . {\displaystyle f:X\to Y.} One may write x ↦ y {\displaystyle x\mapsto y} instead of y = f ( x ) {\displaystyle y=f(x)} , where
3772-468: Is the set of input values that produce y {\displaystyle y} . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ". Similarly,
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3864-620: Is the subset of X {\displaystyle X} defined by f − 1 [ B ] = { x ∈ X : f ( x ) ∈ B } . {\displaystyle f^{-1}[B]=\{x\in X\,:\,f(x)\in B\}.} Other notations include f − 1 ( B ) {\displaystyle f^{-1}(B)} and f − ( B ) . {\displaystyle f^{-}(B).} The inverse image of
3956-469: Is typically the case for functions whose domain is the set of the natural numbers . Such a function is called a sequence , and, in this case the element f n {\displaystyle f_{n}} is called the n th element of the sequence. The index notation can also be used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during
4048-557: Is used, related terms like domain , codomain , injective , continuous have the same meaning as for a function. Inverse image In mathematics , for a function f : X → Y {\displaystyle f:X\to Y} , the image of an input value x {\displaystyle x} is the single output value produced by f {\displaystyle f} when passed x {\displaystyle x} . The preimage of an output value y {\displaystyle y}
4140-449: Is when identical sentences have the same truth-value, yet express different propositions. The sentence “I am a philosopher” could have been spoken by both Socrates and Plato. In both instances, the statement is true, but means something different. These problems are addressed in predicate logic by using a variable for the problematic term, so that “X is a philosopher” can have Socrates or Plato substituted for X, illustrating that “Socrates
4232-442: The graph of the function , a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. 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
4324-692: The Riemann hypothesis . In computability theory , a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem ). A multivariate function , multivariable function , or function of several variables
4416-413: The complex numbers , one talks respectively of a function of several real variables or of a function of several complex variables . There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below. The functional notation requires that a name is given to the function, which, in the case of a unspecified function
4508-448: The inverse image (or preimage ) of a given subset B {\displaystyle B} of the codomain Y {\displaystyle Y} is the set of all elements of X {\displaystyle X} that map to a member of B . {\displaystyle B.} The image of the function f {\displaystyle f} is the set of all output values it may produce, that is,
4600-411: The truth value of them. On the other hand, some signs can be declarative assertions of propositions, without forming a sentence nor even being linguistic (e.g. traffic signs convey definite meaning which is either true or false). Propositions are also spoken of as the content of beliefs and similar intentional attitudes , such as desires, preferences, and hopes. For example, "I desire that I have
4692-420: The zeros of f. This is one of the reasons for which, in mathematical analysis , "a function from X to Y " may refer to a function having a proper subset of X as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers , typically a subset that contains a non-empty open interval . Such
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#17327660939944784-550: The "total" condition removed. That is, a partial function from X to Y is a binary relation R between X and Y such that, for every x ∈ X , {\displaystyle x\in X,} there is at most one y in Y such that ( x , y ) ∈ R . {\displaystyle (x,y)\in R.} Using functional notation, this means that, given x ∈ X , {\displaystyle x\in X,} either f ( x ) {\displaystyle f(x)}
4876-452: The above example), or an expression that can be evaluated to an element of the domain ( x 2 + 1 {\displaystyle x^{2}+1} in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let f ( x ) = sin ( x 2 + 1 ) {\displaystyle f(x)=\sin(x^{2}+1)} ". When
4968-432: The arrow notation for functions described above. In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function f can be defined as mapping any pair of real numbers ( x , y ) {\displaystyle (x,y)} to the sum of their squares, x 2 + y 2 {\displaystyle x^{2}+y^{2}} . Such
5060-590: The arrow notation. The expression x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f ( x 0 , t 0 ) refers to the value of the function f at the point ( x 0 , t 0 ) . Index notation may be used instead of functional notation. That is, instead of writing f ( x ) , one writes f x . {\displaystyle f_{x}.} This
5152-434: The concept of a relation, but using more notation (including set-builder notation ): A function is formed by three sets, the domain X , {\displaystyle X,} the codomain Y , {\displaystyle Y,} and the graph R {\displaystyle R} that satisfy the three following conditions. Partial functions are defined similarly to ordinary functions, with
5244-479: The domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets X and Y is a subset of the set of all ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x\in X} and y ∈ Y . {\displaystyle y\in Y.} The set of all these pairs
5336-435: The domain of f {\displaystyle f} . Throughout, let f : X → Y {\displaystyle f:X\to Y} be a function. The image under f {\displaystyle f} of a subset A {\displaystyle A} of X {\displaystyle X} is the set of all f ( a ) {\displaystyle f(a)} for
5428-410: The domain of definition of a complex function is illustrated by the multiplicative inverse of the Riemann zeta function : the determination of the domain of definition of the function z ↦ 1 / ζ ( z ) {\displaystyle z\mapsto 1/\zeta (z)} is more or less equivalent to the proof or disproof of one of the major open problems in mathematics,
5520-448: The domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse. Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set C {\displaystyle \mathbb {C} } of the complex numbers . The difficulty of determining
5612-489: The fibers over the elements of Y {\displaystyle Y} is a family of sets indexed by Y . {\displaystyle Y.} For example, for the function f ( x ) = x 2 , {\displaystyle f(x)=x^{2},} the inverse image of { 4 } {\displaystyle \{4\}} would be { − 2 , 2 } . {\displaystyle \{-2,2\}.} Again, if there
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#17327660939945704-708: The following properties hold: Also: For functions f : X → Y {\displaystyle f:X\to Y} and g : Y → Z {\displaystyle g:Y\to Z} with subsets A ⊆ X {\displaystyle A\subseteq X} and C ⊆ Z , {\displaystyle C\subseteq Z,} the following properties hold: For function f : X → Y {\displaystyle f:X\to Y} and subsets A , B ⊆ X {\displaystyle A,B\subseteq X} and S , T ⊆ Y , {\displaystyle S,T\subseteq Y,}
5796-420: The following properties hold: The results relating images and preimages to the ( Boolean ) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets: (Here, S {\displaystyle S} can be infinite, even uncountably infinite .) With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism , while
5888-408: The founders of calculus , Leibniz , Newton and Euler . However, it cannot be formalized , since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory . This set-theoretic definition is based on the fact that a function establishes a relation between the elements of
5980-474: The function f (⋅) from its value f ( x ) at x . For example, a ( ⋅ ) 2 {\displaystyle a(\cdot )^{2}} may stand for the function x ↦ a x 2 {\displaystyle x\mapsto ax^{2}} , and ∫ a ( ⋅ ) f ( u ) d u {\textstyle \int _{a}^{\,(\cdot )}f(u)\,du} may stand for
6072-577: The function's domain such that f ( x ) ∈ S . {\displaystyle f(x)\in S.} However, f {\displaystyle f} takes [all] values in S {\displaystyle S} and f {\displaystyle f} is valued in S {\displaystyle S} means that f ( x ) ∈ S {\displaystyle f(x)\in S} for every point x {\displaystyle x} in
6164-407: The function. If the element y in Y is assigned to x in X by the function f , one says that f maps x to y , and this is commonly written y = f ( x ) . {\displaystyle y=f(x).} In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable , and the corresponding element of Y
6256-551: The image of B {\displaystyle B} under f − 1 . {\displaystyle f^{-1}.} The traditional notations used in the previous section do not distinguish the original function f : X → Y {\displaystyle f:X\to Y} from the image-of-sets function f : P ( X ) → P ( Y ) {\displaystyle f:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)} ; likewise they do not distinguish
6348-402: The image of X {\displaystyle X} . The preimage of f {\displaystyle f} , that is, the preimage of Y {\displaystyle Y} under f {\displaystyle f} , always equals X {\displaystyle X} (the domain of f {\displaystyle f} ); therefore, the former notion
6440-593: The inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternative is to give explicit names for the image and preimage as functions between power sets: For every function f : X → Y {\displaystyle f:X\to Y} and all subsets A ⊆ X {\displaystyle A\subseteq X} and B ⊆ Y , {\displaystyle B\subseteq Y,}
6532-436: The notation x ↦ f ( x ) , {\displaystyle x\mapsto f(x),} the symbol x does not represent any value; it is simply a placeholder , meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an interpunct " ⋅ ". This may be useful for distinguishing
6624-404: The partial function is a total function . In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain. In calculus , a real-valued function of a real variable or real function is a partial function from
6716-420: 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, and, 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 , and this greatly increased
6808-627: The possible applications of the concept. A function is often denoted by a letter such as f , g or h . The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x ) is denoted by f ( x ) ; for example, the value of f at x = 4 is denoted by f (4) . Commonly, a specific function is defined by means of an expression depending on x , such as f ( x ) = x 2 + 1 ; {\displaystyle f(x)=x^{2}+1;} in this case, some computation, called function evaluation , may be needed for deducing
6900-401: The power set of X . {\displaystyle X.} The notation f − 1 {\displaystyle f^{-1}} should not be confused with that for inverse function , although it coincides with the usual one for bijections in that the inverse image of B {\displaystyle B} under f {\displaystyle f} is
6992-413: The problem of ambiguity in common language, resulting in a mistaken equivalence of the statements. “I am Spartacus” spoken by Spartacus is the declaration that the individual speaking is called Spartacus and it is true. When spoken by John Smith, it is a declaration about a different speaker and it is false. The term “I” means different things, so “I am Spartacus” means different things. A related problem
7084-415: The proposition that the sky is blue could be modeled as the set { w , w ′ } {\displaystyle \{w,w'\}} . Numerous refinements and alternative notions of proposition-hood have been proposed including inquisitive propositions and structured propositions . Propositions are called structured propositions if they have constituents, in some broad sense. Assuming
7176-461: The same across different thinkers). Philosophical debates surrounding propositions as they relate to propositional attitudes have also recently centered on whether they are internal or external to the agent, or whether they are mind-dependent or mind-independent entities. For more, see the entry on internalism and externalism in philosophy of mind. In modern logic, propositions are standardly understood semantically as indicator functions that take
7268-405: The same proposition, if and only if they mean the same thing. The above definitions can result in two identical sentences/sentence-tokens appearing to have the same meaning, and thus expressing the same proposition and yet having different truth-values, as in "I am Spartacus" said by Spartacus and said by John Smith, and "It is Wednesday" said on a Wednesday and on a Thursday. These examples reflect
7360-426: The same set (the set of all possible worlds). Function (mathematics) In mathematics , a function from a set X 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,
7452-408: The set R {\displaystyle \mathbb {R} } of the real numbers to itself. Given a real function f : x ↦ f ( x ) {\displaystyle f:x\mapsto f(x)} its multiplicative inverse x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} is also a real function. The determination of
7544-685: The set { y ∈ Y : x R y for some x ∈ X } {\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}} is called the image, or the range, of R . {\displaystyle R.} Dually, the set { x ∈ X : x R y for some y ∈ Y } {\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}
7636-803: The set of all n -tuples ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} such that x 1 ∈ X 1 , … , x n ∈ X n {\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}} is called the Cartesian product of X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},} and denoted X 1 × ⋯ × X n . {\displaystyle X_{1}\times \cdots \times X_{n}.} Therefore,
7728-865: The set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation . Commonly, an n -tuple is denoted enclosed between parentheses, such as in ( 1 , 2 , … , n ) . {\displaystyle (1,2,\ldots ,n).} When using functional notation , one usually omits the parentheses surrounding tuples, writing f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} instead of f ( ( x 1 , … , x n ) ) . {\displaystyle f((x_{1},\ldots ,x_{n})).} Given n sets X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},}
7820-432: The sky is green. However, a number of alternative formalizations have been proposed, notably the structured propositions view. Propositions have played a large role throughout the history of logic , linguistics , philosophy of language , and related disciplines. Some researchers have doubted whether a consistent definition of propositionhood is possible, David Lewis even remarking that "the conception we associate with
7912-594: The study of a problem. For example, the map x ↦ f ( x , t ) {\displaystyle x\mapsto f(x,t)} (see above) would be denoted f t {\displaystyle f_{t}} using index notation, if we define the collection of maps f t {\displaystyle f_{t}} by the formula f t ( x ) = f ( x , t ) {\displaystyle f_{t}(x)=f(x,t)} for all x , t ∈ X {\displaystyle x,t\in X} . In
8004-443: The symbol ↦ {\displaystyle \mapsto } (read ' maps to ') is used to specify where a particular element x in the domain is mapped to by f . This allows the definition of a function without naming. For example, the square function is the function x ↦ x 2 . {\displaystyle x\mapsto x^{2}.} The domain and codomain are not always explicitly given when
8096-447: The symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin( x ) . Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case,
8188-427: The use of the term " statement ". In Aristotelian logic a proposition was defined as a particular kind of sentence (a declarative sentence ) that affirms or denies a predicate of a subject , optionally with the help of a copula . Aristotelian propositions take forms like "All men are mortal" and "Socrates is a man." Aristotelian logic identifies a categorical proposition as a sentence which affirms or denies
8280-432: The value of the function at a particular value; for example, if f ( x ) = x 2 + 1 , {\displaystyle f(x)=x^{2}+1,} then f ( 4 ) = 4 2 + 1 = 17. {\displaystyle f(4)=4^{2}+1=17.} Given its domain and its codomain, a function is uniquely represented by the set of all pairs ( x , f ( x )) , called
8372-504: The word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as Serge Lang , use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions. In the theory of dynamical systems , a map denotes an evolution function used to create discrete dynamical systems . See also Poincaré map . Whichever definition of map
8464-403: The word ‘proposition’ may be something of a jumble of conflicting desiderata". The term is often used broadly and has been used to refer to various related concepts. In relation to the mind, propositions are discussed primarily as they fit into propositional attitudes . Propositional attitudes are simply attitudes characteristic of folk psychology (belief, desire, etc.) that one can take toward
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