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Counting
COUNTING is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting
Counting
using tally marks at Hanakapiai Beach Counting
Counting
sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...)
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Combinatorics
COMBINATORICS is a branch of mathematics concerning the study of finite or countable discrete structures . Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization ), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics ). Combinatorial problems arise in many areas of pure mathematics , notably in algebra , probability theory , topology , and geometry , and combinatorics also has many applications in mathematical optimization , computer science , ergodic theory and statistical physics
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Infinite Set
In set theory , an INFINITE SET is a set that is not a finite set . Infinite sets may be countable or uncountable . Some examples are: * the set of all integers , {..., -1, 0, 1, 2, ...}, is a countably infinite set; and * the set of all real numbers is an uncountably infinite set .CONTENTS * 1 Properties * 2 See also * 3 References * 4 External links PROPERTIESThe set of natural numbers (whose existence is postulated by the axiom of infinity ) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset
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Real Number
In mathematics , 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
René Descartes
, who distinguished between real and imaginary roots of polynomials . The real numbers include all the rational numbers , such as the integer −5 and the fraction 4/3, and all the irrational numbers , such as √2 (1.41421356..., the square root of 2 , an irrational algebraic number ). Included within the irrationals are the transcendental numbers , such as π (3.14159265...). Real numbers can be thought of as points on an infinitely long line called the number line or real line , where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation , such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one
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Function Composition
In mathematics , FUNCTION COMPOSITION is the pointwise application of one function to the result of another to produce a third function. For instance, the functions f : X → Y and g : Y → Z can be composed to yield a function which maps x in X to g(f(x)) in Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f ", or "g round f ", or "g composed with f ", "g after f ", "g following f ", or "g of f", or "g on f ". Intuitively, composing two functions is a chaining process in which the output of the inner function becomes the input of the outer function. The composition of functions is a special case of the composition of relations , so all properties of the latter are true of composition of functions. The composition of functions has some additional properties
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One-to-one Correspondence
In mathematics , a BIJECTION, BIJECTIVE FUNCTION or ONE-TO-ONE CORRESPONDENCE is a function 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 f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. 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 number , a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a permutation
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Octave
In music , an OCTAVE (Latin : octavus: eighth) or PERFECT OCTAVE is the interval between one musical pitch and another with half or double its frequency . It is defined by ANSI as the unit of frequency level when the base of the logarithm is two. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems". The most important musical scales are typically written using eight notes, and the interval between the first and last notes is an octave. For example, the C major scale is typically written C D E F G A B C, the initial and final Cs being an octave apart. Two notes separated by an octave have the same letter name and are of the same pitch class . Three commonly cited examples of melodies featuring the perfect octave as their opening interval are "Singin\' in the Rain ", "Over the Rainbow ", and " Stranger on the Shore "
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Pre-math Skills
"Pre-math skills" (referred to in British English
British English
as PRE-MATHS SKILLS) is a term used in some countries to refer to math skills learned by preschoolers and kindergarten students, including learning to count numbers (usually from 1 to 10 but occasionally including 0), learning the proper sequencing of numbers, learning to determine which shapes are bigger or smaller, and learning to count objects on a screen or book
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Uncountable Set
In mathematics , an UNCOUNTABLE SET (or UNCOUNTABLY INFINITE SET) is an infinite set that contains too many elements to be countable . The uncountability of a set is closely related to its cardinal number : a set is uncountable if its cardinal number is larger than that of the set of all natural numbers . CONTENTS * 1 Characterizations * 2 Properties * 3 Examples * 4 Without the axiom of choice * 5 See also * 6 References * 7 External links CHARACTERIZATIONSThere are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds: * There is no injective function from X to the set of natural numbers. * X is nonempty and for every ω-sequence of elements of X, there exist at least one element of X not included in it
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Set Theory
SET THEORY is a branch of mathematical logic that studies sets , which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects . The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory , such as the Russell\'s paradox , numerous axiom systems were proposed in the early twentieth century, of which the Zermelo– Fraenkel axioms , with the axiom of choice , are the best-known. Set theory
Set theory
is commonly employed as a foundational system for mathematics , particularly in the form of Zermelo– Fraenkel set theory with the axiom of choice
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Level Of Measurement
LEVEL OF MEASUREMENT or SCALE OF MEASURE is a classification that describes the nature of information within the values assigned to variables . Psychologist Stanley Smith Stevens developed the best known classification with four levels, or scales, of measurement: nominal , ordinal , interval , and ratio . This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines. Other classifications include those by Mosteller and Tukey, and by Chrisman
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Developmental Psychology
DEVELOPMENTAL PSYCHOLOGY is the scientific study of how and why human beings change over the course of their life. Originally concerned with infants and children , the field has expanded to include adolescence , adult development , aging , and the entire lifespan. This field examines change across three major dimensions: physical development, cognitive development , and socioemotional development. Within these three dimensions are a broad range of topics including motor skills , executive functions , moral understanding , language acquisition , social change , personality , emotional development, self-concept and identity formation . Developmental psychology
Developmental psychology
examines the influences of nature and nurture on the process of human development, and processes of change in context and across time
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Permutation
In mathematics , the notion of PERMUTATION relates to the act of ARRANGING all the members of a set into some sequence or order , or if the set is already ordered, REARRANGING (reordering) its elements, a process called PERMUTING. These differ from combinations , which are selections of some members of a set where order is disregarded. For example, written as tuples , there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. In this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics . Permutations occur, in more or less prominent ways, in almost every area of mathematics
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Pigeonhole Principle
In mathematics , the PIGEONHOLE PRINCIPLE states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. This theorem is exemplified in real life by truisms like "in any group of three gloves there must be at least two left gloves or two right gloves". It is an example of a counting argument . This seemingly obvious statement can be used to demonstrate possibly unexpected results; for example, that there are two people in London
London
who have the same number of hairs on their heads. The first formalization of the idea is believed to have been made by Peter Gustav Lejeune Dirichlet
Peter Gustav Lejeune Dirichlet
in 1834 under the name Schubfachprinzip ("drawer principle" or "shelf principle"). For this reason it is also commonly called DIRICHLET\'S BOX PRINCIPLE or DIRICHLET\'S DRAWER PRINCIPLE
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Injective
In mathematics , an INJECTIVE FUNCTION or INJECTION or ONE-TO-ONE FUNCTION is a function that preserves distinctness : it never maps distinct elements of its domain to the same element of its codomain . In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence (a.k.a. bijective function ), which uniquely maps all elements in both domain and codomain to each other (see figures). An injective non-surjective function (injection, not a bijection ) An injective surjective function (bijection ) A non-injective surjective function (surjection , not a bijection ) Occasionally, an injective function from X to Y is denoted f : X ↣ Y, using an arrow with a barbed tail (U+ 21A3 ↣ RIGHTWARDS ARROW WITH TAIL)
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Surjective
In mathematics , a function f from a set X to a set Y is SURJECTIVE (or ONTO), or a SURJECTION, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x is unique ; the function f may map one or more elements of X to the same element of Y. A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value of f(x) for at least one point x in the domain. The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki , a group of mainly French 20th-century mathematicians who under this pseudonym wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain
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