table of mathematical symbols - Article Index

Encyclopedia - Average: Encyclopedia - Average
In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. The most common method...   » Read the article

Encyclopedia - Mean: Encyclopedia - Mean
In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to di...   » Read the article

Encyclopedia - Mathematics: Encyclopedia - Mathematics
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematic...   » Read the article

Encyclopedia - Cantor's Diagonal Argument: Encyclopedia - Cantor's Diagonal Argument
Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is als...   » Read the article

Encyclopedia - Cantor's Theorem: Encyclopedia - Cantor's Theorem
In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardi...   » Read the article

Encyclopedia - Cantor's Theorem: Encyclopedia Ii - Cantor's Theorem - A Detailed Explanation Of The Proof When X Is Countably Infinite
To get a handle on the proof, let's examine it for the specific case when X is countably infinite. Without loss of generality, we may tak...   » Read the article

Encyclopedia - Cantor's Diagonal Argument: Encyclopedia Ii - Cantor's Diagonal Argument - Real Numbers
Cantor's original proof shows that the interval [0,1] is not countably infinite. The proof by contradiction proceeds as follows: Assume ...   » Read the article

Encyclopedia - Classical Treatment Of Tensors: Encyclopedia Ii - Classical Treatment Of Tensors - Contravariant And Covariant Tensors
A contravariant tensor of order 1(Ti) is defined as: A covariant tensor of order 1(Ti) is defined as: ... See also:Classical trea...   » Read the article

Encyclopedia - Mathematical Notation: Encyclopedia Ii - Mathematical Notation - History
Mathematical notation - Counting. It is believed that a mathematical notation was first developed at least 50,000 years ago in order to...   » Read the article

Encyclopedia - Second-order Logic: Encyclopedia Ii - Second-order Logic - The History And Disputed Value Of Second-order Logic
When predicate logic was introduced to the mathematical community by Frege (and independently — and more influentially — by Peirce, w...   » Read the article

Encyclopedia - Average: Encyclopedia Ii - Average - Arithmetic Mean
The arithmetic mean is the standard "average", often simply called the "mean". It is used for many purposes and may be abused by using it...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Major Themes In Mathematics
An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possib...   » Read the article

Encyclopedia - Intersection Set Theory: Encyclopedia Ii - Intersection Set Theory - Arbitrary Intersections
The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a nonempty set whose elements are themse...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Arithmetic Mean
The arithmetic mean is the "standard" average, often simply called the "mean". The mean may often be confused with the median or mode....   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Notation, Language, And Rigor
Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Inspiration, Pure And Applied Mathematics, And Aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Geometric Mean
The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Notation Language And Rigor
Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words...   » Read the article

Encyclopedia - Classical Treatment Of Tensors: Encyclopedia Ii - Classical Treatment Of Tensors - General Tensors
A multi-order (general) tensor is simply the tensor product of single order tensors: such that: This is sometimes termed the tensor...   » Read the article

Encyclopedia - Second-order Logic: Encyclopedia Ii - Second-order Logic - Why Second-order Logic Is Not Reducible To First-order Logic
An optimist might attempt to reduce second-order logic to first-order logic in the following way. Expand the domain from the set of all r...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Harmonic Mean
The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distan...   » Read the article

Encyclopedia - Second-order Logic: Encyclopedia Ii - Second-order Logic - Second-order Logic And Metalogical Results
It is a corollary of Gödel's incompleteness theorem that one cannot have any notion of provability of second-order formulas given the st...   » Read the article

Encyclopedia - Mathematical Notation: Encyclopedia Ii - Mathematical Notation - Precise Semantic Meaning
Precision is necessary so that we can know what we are investigating. Suppose that we have statements, denoted by some formal sequence of...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Weighted Mean
The weighted mean is used, if one wants to combine average values from samples of the same population with different sample sizes: The...   » Read the article

Encyclopedia - Intersection Set Theory: Encyclopedia Ii - Intersection Set Theory - Nullary Intersection
Note that in the previous section we excluded the case where M was the empty set (∅). The reason is the follows. The intersection of th...   » Read the article

Encyclopedia - Intersection Set Theory: Encyclopedia Ii - Intersection Set Theory - Basic Definition
The intersection of A and B is written "A ∩ B". Formally: x is an element of A ∩ B if and only if x is an element of A and x is an ...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Interquartile Mean
The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the high...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Mean Of A Function
In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over i...   » Read the article

Encyclopedia - Cantor's Theorem: Encyclopedia Ii - Cantor's Theorem - The Proof
Let f be any one-to-one function from A into the power set of A. It must be shown that f is necessarily not surjective. To do that, it is...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - History
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matt...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Inspiration Pure And Applied Mathematics And Aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in...   » Read the article

Encyclopedia - Mean: Encyclopedia Ii - Mean - Generalized Mean
The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the Arithmetic, Geometric and Harmonic Means. ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Common Misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Overview Of Fields Of Mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the re...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Is Mathematics A Science?
Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. If one considers science to be strictly about the physical wor...   » Read the article

Encyclopedia - Average: Encyclopedia Ii - Average - Other Averages
The geometric mean, harmonic mean, generalized mean, weighted mean, truncated mean, and interquartile mean are described in their own art...   » Read the article