part_of measure 
In ___MASK84___ , especially in set theory , a set " A " is a subset of a set " B ", or equivalently " B " is a superset of " A ", if " A " is " contained " inside " B ", that is , all elements of " A " are also elements of " B ". " A " and " B " may coincide . The relationship of one set being a subset of another is called inclusion or sometimes containment . Real analysis ( traditionally , the ___MASK93___ of functions of a real variable ) is a branch of mathematical analysis dealing with the real numbers and real - valued functions of a real variable . In particular , it deals with the analytic properties of real functions and sequences , including convergence and limits of sequences of real numbers , the calculus of the real numbers , and continuity , smoothness and related properties of real - valued functions . In geometric measurements , length is the most extended dimension of an object . In the International ___MASK32___ of Quantities , length is any quantity with dimension distance . In other contexts " length " is the measured dimension of an object . For example , it is possible to cut a length of a wire which is shorter than wire thickness . In ___MASK83___ , a measure on a set is a systematic way to assign a number to each suitable subset of that set , intuitively interpreted as its size . In this sense , a measure is a generalization of the concepts of length , area , and volume . A particularly important example is the Lebesgue measure on a Euclidean space , which assigns the conventional length , area , and volume of ___MASK98___ to suitable subsets of the n - dimensional Euclidean space Rn . For instance , the Lebesgue measure of the interval ( 0 , 1 ) in the real numbers is its length in the everyday sense of the word -- specifically , 1 . Technically , a measure is a function that assigns a non - negative real number or +  to ( certain ) subsets of a set X ( see Definition below ) . It must assign 0 to the empty set and be ( countably ) additive : the measure of a ' large ' subset that can be decomposed into a finite ( or countable ) number of ' smaller ' disjoint subsets , is the sum of the measures of the `` smaller '' subsets . In general , if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure , one only finds trivial examples like the counting measure . This problem was resolved by defining measure only on a sub - collection of all subsets ; the so - called measurable subsets , which are required to form a  - ___MASK43___ . This means that countable unions , countable intersections and complements of measurable subsets are measurable . Non - measurable sets in a Euclidean space , on which the Lebesgue measure can not be defined consistently , are necessarily complicated in the sense of being badly mixed up with their complement . Indeed , their existence is a non - trivial consequence of the axiom of choice . Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel , Henri Lebesgue , Johann Radon , and Maurice Fréchet , among others . The main applications of measures are in the foundations of the Lebesgue integral , in Andrey Kolmogorov ' s axiomatisation of probability theory and in ergodic theory . In integration theory , specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover , the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor , the Riemann integral . Probability theory considers measures that assign to the whole set the size 1 , and considers measurable subsets to be events whose probability is given by the measure . Ergodic theory considers measures that are invariant under , or arise naturally from , a dynamical system . ___MASK98___ is a mathematical ___MASK32___ attributed to the Alexandrian Greek mathematician Euclid , which he described in his textbook on ___MASK97___ : the " Elements ". Euclid ' s method consists in assuming a small set of intuitively appealing axioms , and deducing many other propositions ( theorems ) from these . Although many of Euclid ' s results had been stated by earlier mathematicians , Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical ___MASK32___ . The " Elements " begins with plane ___MASK97___ , still taught in secondary ___MASK85___ as the first axiomatic ___MASK32___ and the first examples of formal proof . It goes on to the ___MASK51___ ___MASK97___ of three dimensions . Much of the " Elements " states results of what are now called algebra and number theory , explained in geometrical language . In ___MASK97___ , Euclidean space encompasses the two - dimensional Euclidean plane , the three - dimensional space of ___MASK98___ , and certain other spaces . It is named after the Ancient Greek mathematician Euclid of Alexandria . The term " Euclidean " distinguishes these spaces from other types of spaces considered in modern ___MASK97___ . Euclidean spaces also generalize to higher dimensions . The International ___MASK32___ of Units ( SI ) specifies a set of seven base units from which all other SI units of measurement are derived . Each of these other units ( SI derived units ) is either dimensionless or can be expressed as a product of ( positive or negative , but usually integral ) powers of one or more of the base units . In ___MASK84___ , a polynomial is an expression consisting of variables ( or indeterminates ) and coefficients , that involves only the operations of addition , subtraction , multiplication , and non - negative integer exponents . An example of a polynomial of a single indeterminate is . An example in three variables is . The term proposition has a broad use in contemporary ___MASK28___ . It is used to refer to some or all of ___MASK23___ : the primary bearers of truth - value , the objects of belief and other " propositional attitudes " ( i . e ., what is believed , doubted , etc .), the referents of that - clauses , and the meanings of declarative sentences . Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity . This stipulation rules out certain candidates for propositions , including ___MASK39___ - and utterance - tokens which are not sharable , and concrete events or facts , which cannot be false . In measure ___MASK93___ , the Lebesgue measure , named after French mathematician Henri Lebesgue , is the standard way of assigning a measure to subsets of " n "- dimensional Euclidean space . For " n " = 1 , 2 , or 3 , it coincides with the standard measure of length , area , or volume . In general , it is also called n "- dimensional volume , n "- volume , or simply volume . It is used throughout real analysis , in particular to define Lebesgue integration . Sets that can be assigned a Lebesgue measure are called Lebesgue measurable ; the measure of the Lebesgue measurable set " A " is denoted by  (" A "). ___MASK83___ is the branch of ___MASK84___ dealing with limits and related theories , such as differentiation , integration , measure , infinite series , and analytic functions . Number ___MASK93___ or , in older usage , arithmetic is a branch of pure ___MASK84___ devoted primarily to the study of the integers . It is sometimes called " The Queen of ___MASK84___ " because of its foundational place in the discipline . Number theorists study prime numbers as well as the properties of objects made out of integers ( e . g ., rational numbers ) or defined as generalizations of the integers ( e . g ., algebraic integers ). Space is the boundless three - dimensional extent in which objects and events have relative position and direction . Physical space is often conceived in three linear dimensions , although modern physicists usually consider it , with time , to be ___MASK33___ of a boundless four - dimensional continuum known as ___MASK45___ . The concept of space is considered to be of fundamental importance to an understanding of the physical universe . However , disagreement continues between philosophers over whether it is itself an entity , a relationship between entities , or ___MASK33___ of a conceptual framework . A wire is a single , usually cylindrical , flexible strand or rod of metal . Wires are used to bear mechanical loads or electricity and telecommunications signals . Wire is commonly formed by ___MASK30___ the metal through a hole in a die or draw plate . Wire gauges come in various standard sizes , as expressed in terms of a gauge number . The term " wire " is also used more loosely to refer to a bundle of such strands , as in " multistranded wire ", which is more correctly termed a wire rope in mechanics , or a cable in electricity . In ___MASK84___ , a theorem is a statement that has been proved on the basis of previously established statements , such as other theorems  and generally accepted statements , such as axioms . A theorem is a logical consequence of the axioms . The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system . The proof of a theorem is often interpreted as justification of the truth of the theorem statement . In light of the requirement that theorems be proved , the concept of a theorem is fundamentally " deductive ", in contrast to the notion of a scientific law , which is " experimental ". Alexandria ( or ; ___MASK95___ : '; '; "") is the second largest city and a major economic centre in Egypt , extending about along the coast of the Mediterranean ___MASK3___ in the north central part of the country . Its low elevation on the Nile delta makes it highly vulnerable to rising ___MASK3___ levels . Alexandria is Egypt ' s largest seaport , serving approximately 80 % of Egypt ' s imports and exports . It is an important industrial center because of its natural gas and oil pipelines from Suez . Alexandria is also an important tourist destination . Paint is any ___MASK75___ , liquefiable , or mastic composition that , after application to a substrate in a thin layer , converts to a ___MASK51___ film . It is most commonly used to protect , color , or provide texture to objects . Paint can be made or purchased in many colors  and in many different types , such as watercolor , synthetic , etc . Paint is typically stored , sold , and applied as a ___MASK75___ , but most types dry into a ___MASK51___ . ___MASK51___ is one of the four fundamental states of matter ( the others being liquid , gas , and plasma ). It is characterized by structural rigidity and resistance to changes of shape or volume . Unlike a liquid , a ___MASK51___ object does not flow to take on the shape of its container , nor does it expand to fill the entire volume available to it like a gas does . The atoms in a ___MASK51___ are tightly bound to each other , either in a regular geometric lattice ( crystalline solids , which include metals and ordinary ice ) or irregularly ( an amorphous ___MASK51___ such as common window glass ). A secondary ___MASK85___ , often referred to as a high ___MASK85___ or a senior high ___MASK85___ , is a ___MASK85___ which provides secondary education , between the ages of 11 and 19 depending on location , after primary ___MASK85___ and before higher education . In ___MASK84___ , an analytic function is a function that is locally given by a convergent power series . There exist both real analytic functions and complex analytic functions , categories that are similar in some ways , but different in others . Functions of each type are infinitely differentiable , but complex analytic functions exhibit properties that do not hold generally for real analytic functions . A function is analytic if and only if its Taylor series about " x " converges to the function in some neighborhood for every " x " in its domain . ___MASK70___ is one of the four fundamental states of matter ( the others being ___MASK51___ , liquid , and plasma ). A pure gas may be made up of individual atoms ( e . g . a noble gas like neon ), elemental molecules made from one type of atom ( e . g . oxygen ), or compound molecules made from a variety of atoms ( e . g . carbon dioxide ). A gas mixture would contain a variety of pure gases much like the air . What distinguishes a gas from liquids and solids is the vast separation of the individual gas particles . This separation usually makes a colorless gas invisible to the human observer . The interaction of gas particles in the presence of electric and gravitational fields are considered negligible as indicated by the constant velocity vectors in the image . One type of commonly known gas is steam . An axiom or postulate is a statement that is taken to be true , to serve as a premise or starting point for further ___MASK31___ and arguments . The word comes from the Greek " axíma " ' that which is ___MASK39___ worthy or fit ' or ' that which commends itself as evident .' Euclid (" Eukleids " ; fl . 300 BCE ), sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara , was a Greek mathematician , often referred to as the " father of ___MASK97___ ". He was active in Alexandria during the reign of Ptolemy I ( 323  283 BCE ). His " Elements " is one of the most influential works in the ___MASK22___ of ___MASK84___ , serving as the main textbook for teaching ___MASK84___ ( especially ___MASK97___ ) from the time of its publication until the late 19th or early 20th century . In the " Elements ", Euclid deduced the principles of what is now called Euclidean ___MASK97___ from a small set of axioms . Euclid also wrote works on perspective , conic sections , spherical ___MASK97___ , number theory and rigor . A ___MASK75___ is a nearly incompressible fluid that conforms to the shape of its container but retains a ( nearly ) constant volume independent of pressure . As such , it is one of the four fundamental states of matter ( the others being ___MASK51___ , gas , and plasma ), and is the only state with a definite volume but no fixed shape . A ___MASK75___ is made up of tiny vibrating particles of matter , such as atoms , held together by intermolecular bonds . Water is , by far , the most common ___MASK75___ on Earth . Like a gas , a ___MASK75___ is able to flow and take the shape of a container . Most liquids resist compression , although others can be compressed . Unlike a gas , a ___MASK75___ does not disperse to fill every space of a container , and maintains a fairly constant density . A distinctive property of the ___MASK75___ state is surface tension , leading to wetting phenomena . In ___MASK84___ , a planar lamina is a closed set in a plane of mass formula_1 and surface density formula_2 such that : The surface area of a ___MASK51___ object is a measure of the total area that the surface of the object occupies . The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one - dimensional curves , or of the surface area for polyhedra ( i . e ., objects with flat polygonal faces ), for which the surface area is the sum of the areas of its faces . Smooth surfaces , such as a sphere , are assigned surface area using their representation as parametric surfaces . This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration . ___MASK84___ ( from Greek  " máthma ",  knowledge , study , learning ) is the study of topics such as quantity ( numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of ___MASK84___ . Volume is the quantity of three - dimensional space enclosed by a closed surface , for example , the space that a substance ( ___MASK51___ , liquid , gas , or plasma ) or shape occupies or contains . Volume is often quantified numerically using the SI derived unit , the cubic metre . The volume of a container is generally understood to be the capacity of the container , i . e . the amount of fluid ( gas or liquid ) that the container could hold , rather than the amount of space the container itself displaces . ___MASK43___ ( from Arabic "" al - jabr "" meaning " reunion of broken parts ") is one of the broad parts of ___MASK84___ , together with number theory , geometry and analysis . In its most general form , algebra is the study of mathematical symbols and the rules for manipulating these symbols ; it is a unifying thread of almost all of ___MASK84___ . As such , it includes everything from elementary equation solving to the study of abstractions such as groups , rings , and fields . The more basic parts of algebra are called elementary algebra , the more abstract parts are called abstract algebra or modern algebra . Elementary algebra is generally considered to be essential for any study of ___MASK84___ , science , or engineering , as well as such applications as medicine and economics . Abstract algebra is a major area in advanced ___MASK84___ , studied primarily by professional mathematicians . In ___MASK84___ , solid geometry is the traditional name for the geometry of three - dimensional Euclidean space . In ___MASK84___ , a real number is a value that represents a quantity along a line . The adjective " real " in this context was introduced in the 17th century by René Descartes , who distinguished between real and imaginary roots of polynomials . ___MASK97___ ( from the ; " geo -" " ___MASK42___ ", "- metron " " measurement ") is a branch of ___MASK84___ concerned with questions of shape , size , relative position of figures , and the properties of space . A mathematician who works in the field of geometry is called a geometer . In ___MASK84___ , an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems . A theory consists of an axiomatic system and all its derived theorems . An axiomatic system that is completely described is a special kind of formal system . A formal theory typically means an axiomatic system , for example formulated within model theory . A formal proof is a complete rendition of a mathematical proof within a formal system . Area is the quantity that expresses the extent of a two - dimensional figure or shape , or planar lamina , in the plane . Surface area is its analog on the two - dimensional surface of a three - dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape , or the amount of paint necessary to cover the surface with a single coat . It is the two - dimensional analog of the length of a ___MASK99___ ( a one - dimensional concept ) or the volume of a ___MASK51___ ( a three - dimensional concept ). Set ___MASK93___ is a branch of ___MASK7___ that studies sets , which informally are collections of objects . Although any type of object can be collected into a set , set ___MASK93___ is applied most often to objects that are relevant to ___MASK84___ . The language of set ___MASK93___ can be used in the definitions of nearly all mathematical objects . René Descartes ( Latinized : Renatus Cartesius ; adjectival form : " Cartesian "; 31 ___MASK36___ 159611 ___MASK91___ 1650 ) was a French philosopher , mathematician , and scientist . Dubbed the father of modern western philosophy , much of subsequent Western philosophy is a response to his writings , which are studied closely to this ___MASK79___ . He spent about 20 years of his life in the Dutch Republic . In ___MASK46___ and ___MASK84___ , the dimension of a mathematical space ( or object ) is informally defined as the minimum number of coordinates needed to specify any point within it . Thus a line has a dimension of one because only one coordinate is needed to specify a point on itfor example , the point at 5 on a number line . A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example , both a latitude and longitude are required to locate a point on the surface of a sphere . The inside of a cube , a cylinder or a sphere is three - dimensional because three coordinates are needed to locate a point within these spaces . Henri Léon Lebesgue ( ___MASK49___ 28 , 1875  July 26 , 1941 ) was a French mathematician most famous for his ___MASK93___ of integration , which was a generalization of the 17th century concept of integration  summing the area between an axis and the curve of a function defined for that axis . His ___MASK93___ was published originally in his ___MASK26___ " Intégrale , longueur , aire " (" Integral , length , area ") at the ___MASK72___ of Nancy during 1902 . Quantity is a property that can exist as a magnitude or multitude . Quantities can be compared in terms of " more ", " less ", or " equal ", or by assigning a numerical value in terms of a unit of measurement . Quantity is among the basic classes of things along with quality , substance , change , and relation . Some quantities are such by their inner ___MASK61___ ( as number ), while others are functioning as states ( properties , dimensions , attributes ) of things such as heavy and light , long and short , broad and narrow , small and great , or much and little . A small quantity is sometimes referred to as a quantulum . The International ___MASK32___ of Quantities ( ISQ ) is a ___MASK32___ based on seven base quantities : length , mass , time , electric current , thermodynamic temperature , amount of substance , and luminous intensity . Other quantities such as area , pressure , and electrical resistance are derived from these base quantities by clear , non - contradictory equations . The ISQ defines the quantities that are measured with the SI units and also includes many other quantities in modern science and technology . The ISQ is defined in the international standard ISO / IEC 80000 , and was finalised in 2009 with the publication of ISO 80000 - 1 .