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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
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

3.
Symmetric relation
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In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. In mathematical notation, this is, ∀ a, b ∈ X is equal to is comparable to, and. are odd, is married to is a fully biological sibling of is a homophone of By definition, a relation cannot be both symmetric and asymmetric. However, a relation can be symmetric nor asymmetric, which is the case for is less than or equal to. Symmetric and antisymmetric are actually independent of other, as these examples show. A symmetric relation that is also transitive and reflexive is an equivalence relation, one way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edges two vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatorially equivalent objects, asymmetric relation Antisymmetric relation Commutative property Symmetry in mathematics Symmetry

4.
Transitive closure
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In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive. Informally, the transitive closure gives you the set of all places you can get to from any starting place. More formally, the closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R. If the binary relation itself is transitive, then the closure is that same binary relation, otherwise. A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitive relations include the equality relation on any set, the less than or equal relation on any ordered set. Symbolically, this can be denoted as, if x < y and y < z then x < z, one example of a non-transitive relation is city x can be reached via a direct flight from city y on the set of all cities. The transitive closure of this relation is a different relation, namely there is a sequence of direct flights that begins at city x, every relation can be extended in a similar way to a transitive relation. An example of a relation with a less meaningful transitive closure is x is the day of the week after y. The transitive closure of this relation is some day x comes after a day y on the calendar, for any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family of transitive relations is again transitive, furthermore, there exists at least one transitive relation containing R, namely the trivial one, X × X. The transitive closure of R is then given by the intersection of all transitive relations containing R, for finite sets, we can construct the transitive closure step by step, starting from R and adding transitive edges. This gives the intuition for a general construction. R ⊆ R +, R + contains all of the R i, so in particular R + contains R. R + is minimal, Let G be any transitive relation containing R, we want to show that R + ⊆ G. It is sufficient to show that for every i >0, R i ⊆ G. Well, and since G is transitive, whenever R i ⊆ G, R i +1 ⊆ G according to the construction of R i and what it means to be transitive. Therefore, by induction, G contains every R i, the intersection of two transitive relations is transitive. The union of two transitive relations need not be transitive, to preserve transitivity, one must take the transitive closure. This occurs, for example, when taking the union of two equivalence relations or two preorders, to obtain a new equivalence relation or preorder one must take the transitive closure