|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{For|the term as used in elementary geometry|congruence (geometry)}}
| | In truth, many CCTV systems are privately owned, designed to protect businesses with large amounts of property like rail yards, scrap yards, warehouses, and department stores. So dvr card 16 channel you can find high speco security 90 off chances that you obtain them just due on the pressure of a salesman or perhaps a salesgirl who's trying to sell CCTV to you personally when you might be in demand for merely a [http://wiki.rh-rpg.pl/index.php?title=Picture_Your_Standalone_Dvr_On_Top._Read_This_And_Make_It_So thief alarm]. Mac dvr viewer software for cctv security cameras Finally the safety DVR is merely that, better because each of the software is loaded inside the DVR as well as doesn't rely on the computers operating system.<br><br> |
|
| |
|
| In [[abstract algebra]], a '''congruence relation''' (or simply '''congruence''') is an [[equivalence relation]] on an [[algebraic structure]] (such as a [[group (mathematics)|group]], [[ring (mathematics)|ring]], or [[vector space]]) that is compatible with the structure. Every congruence relation has a corresponding [[quotient]] structure, whose elements are the [[equivalence class]]es (or '''congruence classes''') for the relation.
| | The Archos DVR Station also permits you to send emails from your TV, playback videos in DVD quality using the S-video, RGB or YPb - Pr video outputs. The CCTV transmits the signal for the DVR where all of the happenings in front of the camera are happening along with the TV display causes it to be viewable.<br><br>There are the ones who ask with an essay or an informative conversation. Now you have countless published articles in circulation that don't have the [http://www.villaespania.se/index.php/Cctv_Surveillance_System_Ppt_Guide quality] had to stir the interest of readers. vols [[http://www.thaitourtalk.com/go.php?http://cctvdvrreviews.com http://www.thaitourtalk.com/go.php?http://cctvdvrreviews.com]] zmodo 16 channel cctv dvr software for mac mini surveillance h.264 dvr Do not repeat the string frequently - which is a spam tactic - but make sure that all of your content ties in directly together with your subject. There is definitely an audience out there for you, but you'll not find them if you dvr cctv software 4 don't get available and show your stuff off. |
| | |
| ==Basic example==
| |
| | |
| The prototypical example of a congruence relation is [[Modular arithmetic#Congruence relation|congruence modulo <math>n</math>]] on the set of [[integer]]s. For a given [[positive integer]] <math>n</math>, two integers <math>a</math> and <math>b</math> are called '''congruent modulo <math>n</math>''', written
| |
| : <math>a \equiv b \pmod{n}</math>
| |
| if <math>a - b</math> is [[divisible]] by <math>n</math> (or equivalently if <math>a</math> and <math>b</math> have the same [[remainder]] when divided by <math>n</math>).
| |
| | |
| for example, <math>37</math> and <math>57</math> are congruent modulo <math>10</math>,
| |
| | |
| : <math>37 \equiv 57 \pmod{10}</math>
| |
| | |
| since <math>37 - 57 = -20</math> is a multiple of 10, or equivalently since both <math>37</math> and <math>57</math> have a remainder of <math>7</math> when divided by <math>10</math>.
| |
| | |
| Congruence modulo <math>n</math> (for a fixed <math>n</math>) is compatible with both [[addition]] and [[multiplication]] on the integers. That is, if
| |
| | |
| : <math>a_1 \equiv a_2 \pmod{n}</math> and <math>b_1 \equiv b_2 \pmod{n}</math>
| |
| | |
| then
| |
| | |
| : <math>a_1 + b_1 \equiv a_2 + b_2 \pmod{n}</math> and <math>a_1 b_1 \equiv a_2b_2 \pmod{n}</math>
| |
| | |
| The corresponding addition and multiplication of equivalence classes is known as [[modular arithmetic]]. From the point of view of abstract algebra, congruence modulo <math>n</math> is a congruence relation on the [[ring (mathematics)|ring]] of integers, and arithmetic modulo <math>n</math> occurs on the corresponding [[quotient ring]].
| |
| | |
| ==Definition==
| |
| The definition of a congruence depends on the type of [[algebraic structure]] under consideration. Particular definitions of congruence can be made for [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[vector space]]s, [[module (mathematics)|modules]], [[semigroup]]s, [[lattice (order)|lattices]], and so forth. The common theme is that a congruence is an [[equivalence relation]] on an algebraic object that is compatible with the algebraic structure, in the sense that the [[Operation (mathematics)|operations]] are [[well-defined]] on the [[equivalence classes]].
| |
| | |
| For example, a group is an algebraic object consisting of a [[set (mathematics)|set]] together with a single [[binary operation]], satisfying certain axioms. If <math>G</math> is a group with operation ∗, a '''congruence relation''' on ''G'' is an equivalence relation ≡ on the elements of ''G'' satisfying
| |
| :''g''<sub>1</sub> ≡ ''g''<sub>2</sub> and ''h''<sub>1</sub> ≡ ''h''<sub>2</sub> ⇒ ''g''<sub>1</sub> ∗ ''h''<sub>1</sub> ≡ ''g''<sub>2</sub> ∗ ''h''<sub>2</sub> | |
| for all ''g''<sub>1</sub>, ''g''<sub>2</sub>, ''h''<sub>1</sub>, ''h''<sub>2</sub> ∈ ''G''. For a congruence on a group, the equivalence class containing the [[identity element]] is always a [[normal subgroup]], and the other equivalence classes are the [[coset]]s of this subgroup. Together, these equivalence classes are the elements of a [[quotient group]].
| |
| | |
| When an algebraic structure includes more than one operation, congruence relations are required to be compatible with each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on a ring must satisfy
| |
| :''r''<sub>1</sub> + ''s''<sub>1</sub> ≡ ''r''<sub>2</sub> + ''s''<sub>2</sub> and ''r''<sub>1</sub>''s''<sub>1</sub> ≡ ''r''<sub>2</sub>''s''<sub>2</sub>
| |
| whenever ''r''<sub>1</sub> ≡ ''r''<sub>2</sub> and ''s''<sub>1</sub> ≡ ''s''<sub>2</sub>. For a congruence on a ring, the equivalence class containing 0 is always a two-sided [[Ideal (ring theory)|ideal]], and the two operations on the set of equivalence classes define the corresponding [[quotient ring]].
| |
| | |
| The general notion of a congruence relation can be given a formal definition in the context of [[universal algebra]], a field which studies ideas common to all [[algebraic structures]]. In this setting, a congruence relation is an equivalence relation ≡ on an algebraic structure that satisfies
| |
| :''μ''(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>) ≡ ''μ''(''a''<sub>1</sub>′, ''a''<sub>2</sub>′, ..., ''a''<sub>''n''</sub>′) | |
| for every ''n''-ary operation ''μ'', and all elements ''a''<sub>1</sub>,...,''a''<sub>''n''</sub>,''a''<sub>1</sub>′,...,''a''<sub>''n''</sub>′ satisfying ''a''<sub>''i''</sub> ≡ ''a''<sub>''i''</sub>′ for each ''i''.
| |
| | |
| ==Relation with homomorphisms==
| |
| If ƒ: ''A'' → ''B'' is a [[homomorphism]] between two algebraic structures (such as [[group homomorphism|homomorphism of groups]], or a [[linear map]] between [[vector space]]s), then the relation ≡ defined by
| |
| :''a''<sub>1</sub> ≡ ''a''<sub>2</sub> if and only if ƒ(''a''<sub>1</sub>) = ƒ(''a''<sub>2</sub>)
| |
| is a congruence relation. By the [[first isomorphism theorem]], the [[image (mathematics)|image]] of ''A'' under ƒ is a substructure of ''B'' [[Isomorphism|isomorphic]] to the quotient of ''A'' by this congruence.
| |
| | |
| ==Congruences of groups, and normal subgroups and ideals==
| |
| In the particular case of [[group (mathematics)|groups]], congruence relations can be described in elementary terms as follows:
| |
| If ''G'' is a group (with [[identity element]] ''e'' and operation *) and ~ is a [[binary relation]] on ''G'', then ~ is a congruence whenever:
| |
| #[[Given any]] element ''a'' of ''G'', ''a'' ~ ''a'' ('''[[Reflexive relation|reflexivity]]''');
| |
| #Given any elements ''a'' and ''b'' of ''G'', [[material conditional|if]] ''a'' ~ ''b'', then ''b'' ~ ''a'' ('''[[Symmetric relation|symmetry]]''');
| |
| #Given any elements ''a'', ''b'', and ''c'' of ''G'', if ''a'' ~ ''b'' [[logical conjunction|and]] ''b'' ~ ''c'', then ''a'' ~ ''c'' ('''[[Transitive relation|transitivity]]''');
| |
| #Given any elements ''a'', ''a' '', ''b'', and ''b' '' of ''G'', if ''a'' ~ ''a' '' and ''b'' ~ ''b' '', then ''a'' * ''b'' ~ ''a' '' * ''b' '';
| |
| #Given any elements ''a'' and ''a' '' of ''G'', if ''a'' ~ ''a' '', then ''a''<sup>−1</sup> ~ ''a' ''<sup>−1</sup> (this can actually be proven from the other four, so is strictly redundant).
| |
| | |
| Conditions 1, 2, and 3 say that ~ is an [[equivalence relation]].
| |
| | |
| A congruence ~ is determined entirely by the set {''a'' ∈ ''G'' : ''a'' ~ ''e''} of those elements of ''G'' that are congruent to the identity element, and this set is a [[normal subgroup]].
| |
| Specifically, ''a'' ~ ''b'' if and only if ''b''<sup>−1</sup> * ''a'' ~ ''e''.
| |
| So instead of talking about congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruence corresponds uniquely to some normal subgroup of ''G''.
| |
| | |
| === Ideals of rings and the general case ===
| |
| | |
| A similar trick allows one to speak of kernels in [[ring (mathematics)|ring theory]] as [[ideal (ring theory)|ideals]] instead of congruence relations, and in [[module (mathematics)|module theory]] as [[submodule]]s instead of congruence relations.
| |
| | |
| The most general situation where this trick is possible is with [[Omega-group]]s (in the general sense allowing operators with multiple arity). But this cannot be done with, for example, [[monoid]]s, so the study of congruence relations plays a more central role in monoid theory.
| |
| | |
| ==Universal algebra==
| |
| | |
| The idea is generalized in [[universal algebra]]:
| |
| A congruence relation on an algebra ''A'' is a [[subset]] of the [[direct product]] ''A'' × ''A'' that is both an [[equivalence relation]] on ''A'' and a [[subalgebra]] of ''A'' × ''A''.
| |
| | |
| The [[kernel (universal algebra)|kernel]] of a [[homomorphism]] is always a congruence. Indeed, every congruence arises as a kernel.
| |
| For a given congruence ~ on ''A'', the set ''A''/~ of [[equivalence class]]es can be given the structure of an algebra in a natural fashion, the [[quotient algebra]].
| |
| The function that maps every element of ''A'' to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.
| |
| | |
| The [[lattice (order)|lattice]] '''Con'''(''A'') of all congruence relations on an algebra ''A'' is [[algebraic lattice|algebraic]].
| |
| | |
| ==See also==
| |
| *[[Table of congruences]]
| |
| *[[Linear congruence theorem]]
| |
| *[[Congruence lattice problem]]
| |
| | |
| ==References==
| |
| * Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5 discusses congruency of matrices.)
| |
| | |
| {{DEFAULTSORT:Congruence Relation}}
| |
| [[Category:Modular arithmetic]]
| |
| [[Category:Algebra]]
| |
| [[Category:Mathematical relations]]
| |
In truth, many CCTV systems are privately owned, designed to protect businesses with large amounts of property like rail yards, scrap yards, warehouses, and department stores. So dvr card 16 channel you can find high speco security 90 off chances that you obtain them just due on the pressure of a salesman or perhaps a salesgirl who's trying to sell CCTV to you personally when you might be in demand for merely a thief alarm. Mac dvr viewer software for cctv security cameras Finally the safety DVR is merely that, better because each of the software is loaded inside the DVR as well as doesn't rely on the computers operating system.
The Archos DVR Station also permits you to send emails from your TV, playback videos in DVD quality using the S-video, RGB or YPb - Pr video outputs. The CCTV transmits the signal for the DVR where all of the happenings in front of the camera are happening along with the TV display causes it to be viewable.
There are the ones who ask with an essay or an informative conversation. Now you have countless published articles in circulation that don't have the quality had to stir the interest of readers. vols [http://www.thaitourtalk.com/go.php?http://cctvdvrreviews.com] zmodo 16 channel cctv dvr software for mac mini surveillance h.264 dvr Do not repeat the string frequently - which is a spam tactic - but make sure that all of your content ties in directly together with your subject. There is definitely an audience out there for you, but you'll not find them if you dvr cctv software 4 don't get available and show your stuff off.