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empty set | A Wisdom Archive on empty set | | empty set A selection of articles related to empty set | |
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empty set
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| ARTICLES RELATED TO empty set | | | | | | | | | | | | |  empty set: Encyclopedia II - Empty product - Nullary arithmetic product
Empty product - Frequent examples.
Two often-seen instances are a0 = 1 (any number raised to the zeroth power is one) and 0! = 1 (the factorial of zero is one). It can also be motivated by the fact that if all factors of the numerator or the denominator in a fraction cancel (as would 2 and 3 in the following example), the remaining value is 1,
The numerator becomes here a "pro ...
See also:Empty product, Empty product - Nullary arithmetic product, Empty product - Frequent examples, Empty product - Conceptual justification, Empty product - Technical justification, Empty product - 0 raised to the 0th power, Empty product - Nullary intersection, Empty product - Nullary categorical product, Empty product - In computer programming, Empty product - Quote Read more here: » Empty product: Encyclopedia II - Empty product - Nullary arithmetic product |
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| | | | | empty set: Encyclopedia II - Order theory - Introduction to the basic definitions This section aims at giving a first guide to the realm of ordered sets. It addresses readers who have basic knowledge of set theory and arithmetics and who know what a binary relation is, but who are not familiar with order theoretic considerations so far.
Order theory - Partially ordered sets.
As already hinted at above, orders are special binary relations. Hence consider some set P and a relation ≤ on P. Then ≤ is a partial order if it is reflexive, antisymmetric, and transitive, ...
See also:Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature Read more here: » Order theory: Encyclopedia II - Order theory - Introduction to the basic definitions |
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| | | | | empty set: Encyclopedia II - Discrete space - Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one another. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space X := {1/n : n = 1,2,3,...} (with metric inherited from the real line and given by d(x,y) = |x − ySee also: Discrete space, Discrete space - Definitions, Discrete space - Properties, Discrete space - Uses, Discrete space - Indiscrete spaces, Discrete space - Quotation, Discrete space - Notes Read more here: » Discrete space: Encyclopedia II - Discrete space - Properties |
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| | | | | empty set: Encyclopedia II - Vacuous truth - Arguments of the semantic truth of vacuously true logical statements This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i.e., the case when S has the form P ⇒ Q, and P is false. This case strikes many people as odd, and it's not immediately obvious whether all such statements are true, all such statements are false, some are true while others are false, or what.
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See also:Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics Read more here: » Vacuous truth: Encyclopedia II - Vacuous truth - Arguments of the semantic truth of vacuously true logical statements |
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| | | | | empty set: Encyclopedia II - Event probability theory - A simple example If we assemble a deck of 52 playing cards and two jokers, and draw a single card from the deck, then the sample space is a 54-element set, as each individual card is a possible outcome. An event, however, is any subset of the sample space, including any single-element set (an elementary event, of which there are 54, representing the 54 possible cards drawn from the deck), the empty set (which is defined to have probability zero) and the entire set of 54 cards, the sample space itself (which is defined to have probability one). Other events a ...
See also:Event probability theory, Event probability theory - A simple example, Event probability theory - Events in probability spaces Read more here: » Event probability theory: Encyclopedia II - Event probability theory - A simple example |
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| | | | | empty set: Encyclopedia II - Open set - Definitions The concept of open sets can be formalized in various degrees of generality.
Open set - Function-analytic.
A point set in Rn is called open when every point P of the set is an inner point.
Open set - Euclidean space.
A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in < ...
See also:Open set, Open set - Definitions, Open set - Function-analytic, Open set - Euclidean space, Open set - Metric spaces, Open set - Topological spaces, Open set - Uses, Open set - Manifolds Read more here: » Open set: Encyclopedia II - Open set - Definitions |
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| | | | | | | empty set: Encyclopedia II - Measure mathematics - Generalizations For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. A measure that takes values in a Banach space is called a spectral measure; these are used mainly in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values fro ...
See also:Measure mathematics, Measure mathematics - Formal definition, Measure mathematics - Properties, Measure mathematics - Monotonicity, Measure mathematics - Measures of infinite unions of measurable sets, Measure mathematics - Measures of infinite intersections of measurable sets, Measure mathematics - Sigma-finite measures, Measure mathematics - Completeness, Measure mathematics - Examples, Measure mathematics - Counterexamples, Measure mathematics - Generalizations Read more here: » Measure mathematics: Encyclopedia II - Measure mathematics - Generalizations |
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| | | | | empty set: Encyclopedia II - Directed set - Directed subsets Directed sets need not be antisymmetric and therefore in general are not partial orders. However, the term is also frequently used in the context of posets. In this setting, a subset A of a partially ordered set (P,≤) is called a directed subset iff
A is not the empty set,
for any two a and b in A, there exists a c in A with a ≤ c and b ≤ c (directedness),
where the order of the elements of A is inherited from P. For this reason, reflexivity ...
See also:Directed set, Directed set - Applications, Directed set - Examples, Directed set - Directed subsets Read more here: » Directed set: Encyclopedia II - Directed set - Directed subsets |
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| | | | | | empty set: Encyclopedia II - Ordinal number - Arithmetic of ordinals To define the sum S + T of two ordinal numbers S and T, one proceeds as follows: first the elements of T are relabeled so that S and T become disjoint, then the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S∪T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This way, a new well-ordered set is formed, ...
See also:Ordinal number, Ordinal number - Introduction, Ordinal number - The oldest definition, Ordinal number - Modern definition and first properties, Ordinal number - Other definitions, Ordinal number - Arithmetic of ordinals, Ordinal number - Cantor normal form, Ordinal number - Topology and limit ordinals Read more here: » Ordinal number: Encyclopedia II - Ordinal number - Arithmetic of ordinals |
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