1.
Truth-value
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In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth. In classical logic, with its intended semantics, the values are true and untrue or false. This set of two values is called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables, logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgans laws, assigning values for propositional variables is referred to as valuation. In intuitionistic logic, and more generally, constructive mathematics, statements are assigned a value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if you can build a proof of the statement from those axioms, a statement is false if you can deduce a contradiction from it. This leaves open the possibility of statements that have not yet assigned a truth value. Unproven statements in Intuitionistic logic are not given a truth value. Indeed, you can prove that they have no truth value. There are various ways of interpreting Intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation, see also, Intuitionistic Logic - Semantics. Multi-valued logics allow for more than two values, possibly containing some internal structure. For example, on the interval such structure is a total order. Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions, but even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, Intuitionistic type theory uses types in the place of truth values. Topos theory uses truth values in a sense, the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational

2.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain

3.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as

4.
Indicator function
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It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript describing the set. The indicator function of a subset A of a set X is a function 1 A, X → defined as 1 A, = {1 if x ∈ A,0 if x ∉ A. The Iverson bracket allows the equivalent notation, to be used instead of 1 A, the function 1 A is sometimes denoted I A, χ A, KA or even just A. The set of all functions on X can be identified with P. This is a case of the notation Y X for the set of all functions f, X → Y. The notation 1 A is also used to denote the identity function of A, the notation χ A is also used to denote the characteristic function in convex analysis. A related concept in statistics is that of a dummy variable, the term characteristic function has an unrelated meaning in probability theory. The indicator or characteristic function of a subset A of some set X and this mapping is surjective only when A is a non-empty proper subset of X. By a similar argument, if A ≡ Ø then 1A =0, in the following, the dot represents multiplication, 1·1 =1, 1·0 =0 etc. + and − represent addition and subtraction. ∩ and ∪ is intersection and union, respectively. More generally, suppose A1, …, A n is a collection of subsets of X, for any x ∈ X, ∏ k ∈ I is clearly a product of 0s and 1s. This product has the value 1 at precisely those x ∈ X that belong to none of the sets Ak and is 0 otherwise and that is ∏ k ∈ I =1 X − ⋃ k A k =1 −1 ⋃ k A k. This is one form of the principle of inclusion-exclusion, as suggested by the previous example, the indicator function is a useful notational device in combinatorics. This identity is used in a proof of Markovs inequality. In many cases, such as theory, the inverse of the indicator function may be defined. This is commonly called the generalized Möbius function, as a generalization of the inverse of the function in elementary number theory. Given a probability space with A ∈ F, the random variable 1 A, Ω → R is defined by 1 A =1 if ω ∈ A. Mean E = P Variance Var = P Covariance Cov = P − P P Kurt Gödel described the function in his 1934 paper On Undecidable Propositions of Formal Mathematical Systems. There shall correspond to each class or relation R a representing function φ =0 if R and φ =1 if ~R, for example, because the product of characteristic functions φ1*φ2*

5.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times