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Counting
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
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Combinatorics
Combinatorics
Combinatorics
is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. To fully understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon.[1] According to H. J
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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.[1] Psychologist Stanley Smith Stevens developed the best known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio.[1][2] This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines.[3] Other classifications include those by Mosteller and Tukey,[4] and by Chrisman.[5]Contents1 Stevens's typology1.1 Overview1.1.1 Comparison1.2 Nominal level1.2.1 Mathematical operations 1.2.2 Central tendency<
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Developmental Psychology
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. Developmental psychologists aim to explain how thinking, feeling and behaviour change throughout life
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Pre-math Skills
"Pre-math skills" (referred to in 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. Pre-math skills are also tied into literacy skills to learn the correct pronunciations of numbers. External links[edit]Pre-math Skills on Bella OnlineThis article relating to education is a stub
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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[2] 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".[3] 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
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Interval (music)
In music theory, an interval is the difference between two pitches.[1] An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.[2][3] In Western music, intervals are most commonly differences between notes of a diatonic scale. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C♯ and D♭. Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1
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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.[note 1] The notation g ∘ f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", or "g on f "
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Element (mathematics)
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.Contents1 Sets 2 Notation and terminology 3 Cardinality
Cardinality
of sets 4 Examples 5 References 6 Further reading 7 External linksSets[edit] Writing A = 1 , 2 , 3 , 4 displaystyle A= 1,2,3,4 means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example 1 , 2 displaystyle 1,2 , are subsets of A. Sets can themselves be elements. For example, consider the set B = 1 , 2 , 3 , 4 displaystyle B= 1,2, 3,4 . The elements of B are not 1, 2, 3, and 4
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Phalanges
The phalanges /fəˈlændʒiːz/ (singular: phalanx /ˈfælæŋks/) are digital bones in the hands and feet of most vertebrates. In primates, the thumbs and big toes have two phalanges while the other digits have three phalanges. The phalanges are classed as long bones.Contents1 Structure1.1 Bone
Bone
anatomy1.1.1 Distal phalanx1.2 Development2 Function 3 History of phalanges3.1 In animals3.1.1 Primates 3.1.2 Other mammals4 Additional images 5 See also 6 References 7 External linksStructure[edit]Bones of footThe phalanges are the bones that make up the fingers of the hand and the toes of the foot. There are 56 phalanges in the human body, with fourteen on each hand and foot. Three phalanges are present on each finger and toe, with the exception of the thumb and large toe, which possess only two
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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.Contents1 Properties 2 See also 3 References 4 External linksProperties[edit] The 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
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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, 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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Uncountable Set
In mathematics, an uncountable set (or uncountably infinite set)[1] 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.Contents1 Characterizations 2 Properties 3 Examples 4 Without the axiom of choice 5 See also 6 References 7 External linksCharacterizations[edit] There 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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Mathematical Induction
Mathematical induction
Mathematical induction
is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for all natural numbers 0, 1, 2, 3, and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder: Mathematical induction
Mathematical induction
proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). — Concrete Mathematics, page 3 margins.The method of induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n+1
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Writing
Writing
Writing
is a medium of human communication that represents language and emotion with signs and symbols. In most languages, writing is a complement to speech or spoken language. Writing
Writing
is not a language, but a tool used to make languages be read. Within a language system, writing relies on many of the same structures as speech, such as vocabulary, grammar, and semantics, with the added dependency of a system of signs or symbols. The result of writing is called text, and the recipient of text is called a reader. Motivations for writing include publication, storytelling, correspondence, record keeping and diary
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Set Theory
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
Georg Cantor
and Richard Dedekind
Richard Dedekind
in the 1870s
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