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| In [[mathematics]], an '''empty product''', or '''nullary product''', is the result of [[multiplication|multiplying]] no factors. It is by convention equal to the multiplicative [[identity element|identity]] [[1 (number)|1]], given that there is an identity for the multiplication operation in question, just as the [[empty sum]]—the result of [[addition|adding]] no numbers—is by convention [[0 (number)|zero]], or the additive identity.<ref>{{cite book |author=Jaroslav Nešetřil, [[Jiří Matoušek (mathematician)|Jiří Matoušek]] |title=Invitation to Discrete Mathematics |publisher=Oxford University Press |year=1998 |isbn=0-19-850207-9 |pages=12}}</ref><ref>{{cite book |author=A.E. Ingham and R C Vaughan |title=The Distribution of Prime Numbers |publisher=Cambridge University Press |year=1990 |isbn=0-521-39789-8 |pages=1}}</ref><ref>Page 9 of {{Lang Algebra|edition=3r}}</ref>
| | Most individuals have this habit of doing all stuff by themselves, regardless of how critical or simple they are! These people won't let others interfere in their affairs. While this stance might work in other regions of life, it's not how to respond when you need to fix your Windows registry. There are some jobs such as removing spywares, virus plus obsolete registry entries, that are best left to specialist softwares. In this particular article I usually tell we why it happens to be critical to fix Windows registry NOW!<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents and difficulties with the stored browsing information. Delete the browsing information and obvious the contents of the cache to resolve this problem.<br><br>System tray icon makes it easy to launch the system and displays "clean" status or the amount of mistakes in the last scan. The ability to locate and remove the Invalid class keys and shell extensions is regarded as the primary advantages of the system. That is not normal function for the additional Registry Cleaners. Class keys plus shell extensions which are not working can really slow down a computer. RegCure scans to obtain invalid entries and delete them.<br><br>Chrome enables customizing itself by applying range of themes available online. If you had lately used a theme that no longer works correctly, it results inside Chrome crash on Windows 7. It is recommended to set the authentic theme.<br><br>Google Chrome crashes on Windows 7 if the registry entries are improperly modified. Missing registry keys or registry keys with wrong values can cause runtime mistakes and thereby the issue happens. We are recommended to scan the entire system registry plus review the result. Attempt the registry repair task using third-party [http://bestregistrycleanerfix.com/registry-mechanic pc tools registry mechanic] software.<br><br>Windows relies heavily on this database, storing everything from the newest emails to the Internet favorites inside there. Because it's thus crucial, the computer is continually adding plus updating the files inside it. This is ok, nevertheless it can create a computer run slow, whenever the computer accidentally breaks its crucial registry files. This is a really usual issue, and actually makes a computer run slower each day. What arises is that since a computer is frequently using 100's of registry files at once, it occasionally gets confused plus make a few of them unreadable. This then makes the computer run slow, because Windows takes longer to read the files it demands.<br><br>By restoring the state of the system to an earlier date, error 1721 could not appear inside Windows 7, Vista plus XP. There is a tool called System Restore that we have to employ inside this procedure.<br><br>You are able to click here to obtain out how to accelerate Windows and grow PC perfomance. And you are able to click here to download a registry cleaner to aid you clean up registry. |
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| When a mathematical recipe says "multiply all the numbers in this list", and the list contains, say, 2, 3, 2 and 4, we multiply first the first number by the second, then the result by the third, and so on until the end of the list, so the product of (2,3,2,4) would be 48. If the list contains only one number, so that we cannot multiply first by second, common convention holds that the 'product of all' is that same number, and if the list has no numbers at all, the 'product of all' is 1. This value is necessary to be consistent with the [[Recursion|recursive]] definition of what a product over a sequence means. For example,
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| : <math>
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| \begin{align}
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| \text{prod}(\{2,3,5\}) & = \text{prod}(\{2,3\}) \times 5 = \text{prod}(\{2\}) \times 3 \times 5 \\
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| & = \text{prod}(\{\}) \times 2 \times 3 \times 5 = 1 \times 2 \times 3 \times 5.
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| \end{align}
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| </math>
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| In general, we define | |
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| : <math>\text{prod}(\{\}) = 1 \qquad \text{prod}(\{a_i\}_{i \le n}) = \text{prod}(\{a_i\}_{i \le n-1}) \times a_n.</math>
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| The empty product is used in [[discrete mathematics]], [[elementary algebra|algebra]], the study of [[power series]], and [[computer programming|computer programs]].
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| The term "empty product" is most often used in the above sense when discussing [[arithmetic]] operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming; these are discussed below.
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| == Nullary arithmetic product ==
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| [[File:Lattice of the divisibility of 60; factors.svg|thumb|[[Lattice (order)|Lattice]] of [[divisor]]s of 60<br>The vertex without prime factors is 1.]]
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| ===Intuitive justification===
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| Imagine a [[calculator]] that can only multiply.
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| It has an "ENTER" key and a "CLEAR" key.
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| One would wish that, for example, if one presses "CLEAR", 7 "ENTER", 3 "ENTER", 4 "ENTER", then the display reads 84, because 7 × 3 × 4 = 84. More precisely, we specify:
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| *A number is displayed just after "CLEAR" is pressed.
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| *When a number is displayed and one enters another number, their product is displayed.
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| Then the starting value after pressing "CLEAR" has to be 1. After one has pressed "clear" and done nothing else, the number of factors one has entered is zero. Therefore the product of zero numbers is 1.
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| ===Frequent examples===
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| Two often-seen instances are ''a''<sup>0</sup> = 1 (any number raised to the zeroth [[exponentiation|power]] is one) and 0! = 1 (the [[factorial]] of zero is one).
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| More examples of the use of the empty product in mathematics may be found in the [[binomial theorem]], [[factorial]], [[fundamental theorem of arithmetic]], [[birthday paradox]], [[Stirling number]], [[König's theorem (set theory)|König's theorem]], [[binomial type]], [[difference operator]], [[Pochhammer symbol]], [[proof that e is irrational]], [[prime factor]],<ref>{{cite web |url=http://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EWD1073.html |title=How Computing Science created a new mathematical style |author=[[Edsger Wybe Dijkstra]] |date=1990-03-04 |work=EWD |accessdate=2010-01-20 | quote=Hardy and Wright: “Every positive integer, except 1, is a product of primes”, Harold M. Stark: “If n is an integer greater than 1, then either n is prime or n is a finite product of primes.”. These examples —which I owe to A.J.M. van Gasteren— both reject the empty product, the last one also rejects the product with a single factor.}}</ref><ref>{{cite web |url=http://userweb.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD993.html |title=The nature of my research and why I do it |author=[[Edsger Wybe Dijkstra]] |date=1986-11-14 |work=EWD |accessdate=2010-07-03 | quote=But also 0 is certainly finite and by defining the product of 0 factors —how else?— to be equal to 1 we can do away with the exception: "If n is a positive integer, then n is a finite product of primes."}}</ref> [[binomial series]], and [[multiset]].
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| ===Logarithms===
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| The definition of an empty product can be based on that of the [[empty sum]]:
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| The sum of two [[logarithm]]s is equal to the logarithm of the product of their operands, i.e. for any base ''b'' > 0:
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| :<math>\log_b n + \log_b m = \log_b (nm) \,</math>
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| and
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| :<math>b^{\log_b n + \log_b m} = nm</math>
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| and more generally
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| :<math>\prod_i x_i = b^{\sum_i \log_b x_i}</math>
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| i.e., multiplication across all elements of a set is ''b'' to the power of the sum of all logarithms of the set's elements.
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| Using this property as definition, and extending this to the empty product, the [[right-hand side]] of this equation evaluates to ''b''<sup>0</sup> for the [[empty set]], because the empty sum is defined to be zero, and therefore the empty product must equal one.
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| == 0 raised to the 0th power ==
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| {{see|Exponentiation#Zero to the power of zero}}
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| In set theory and combinatorics, the [[cardinal number]] ''n<sup>m</sup>'' is the size of the set of functions from a set of size ''m'' into a set of size ''n''. If ''m'' is positive and ''n'' is zero, then there are no such functions, because there are no elements in the set of size 0 to map elements of the set of size m into. Thus 0<sup>''m''</sup> = 0 when ''m'' is positive. However, if both sets are empty (have size 0), then there is exactly one such function — the [[empty function]]. For this reason, authors in combinatorics and set theory frequently define 0<sup>0</sup> to be 1 when it represents an empty product.
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| == Nullary conjunction and intersection ==
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| [[File:Multigrade operator AND.svg|thumb|[[Logical conjunction|Conjunctions]] of the arguments in parentheses: The conjunction of no argument is the [[tautology (logic)|tautology]].]]
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| For similar reasons, the [[logical conjunction]] of no argument is the [[tautology (logic)|tautology]]. Accordingly the [[intersection (set theory)|intersection]] of no set is conventionally equal to the [[universe (set theory)|universe]]. See [[nullary intersection]] for more information.
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| == Nullary Cartesian product ==
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| Consider the general definition of the [[Cartesian product]]:
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| :<math>\prod_{i \in I} X_i = \{ g : I \to \bigcup_{i \in I} X_i\ |\ \forall i\ g(i) \in X_i \}.</math>
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| If ''I'' is empty, the only such ''g'' is the [[empty function]] <math>f_\varnothing</math>, which is the unique subset of <math>\varnothing\times\varnothing</math> that is a function <math>\varnothing \to \varnothing</math>, namely the empty subset <math>\varnothing</math> (the only subset that <math>\varnothing\times\varnothing = \varnothing</math> has):
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| :<math>\prod_\varnothing{} = \{ f_\varnothing: \varnothing \to \varnothing \} = \{ \varnothing\}.</math>
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| Thus, the cardinality of the Cartesian product of no sets is 1.
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| Under the perhaps more familiar ''n''-[[tuple]] interpretation,
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| :<math>\prod_\varnothing{} = \{ ( ) \},</math>
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| that is, the [[singleton set]] containing the [[empty tuple]]. Note that in both representations the empty product has [[cardinality]] 1.
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| === Nullary Cartesian product of functions ===
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| The empty [[Cartesian product#Cartesian product of functions|Cartesian product of functions]] is again the empty function.
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| == Nullary categorical product ==
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| In any [[category (category theory)|category]], the [[product (category theory)|product]] of an empty family is a [[terminal object]] of that category. This can be demonstrated by using the [[limit (category theory)|limit]] definition of the product. An ''n''-fold categorical product can be defined as the limit with respect to a [[diagram (category theory)|diagram]] given by the [[discrete category]] with ''n'' objects. An empty product is then given by the limit with respect to the empty category, which is the terminal object of the category if it exists. This definition specializes to give results as above. For example, in the [[category of sets]] the categorical product is the usual Cartesian product, and the terminal object is a singleton set. In the [[category of groups]] the categorical product is the Cartesian product of groups, and the terminal object is a trivial group with one element. To obtain the usual arithmetic definition of the empty product we must take the [[decategorification]] of the empty product in the category of finite sets.
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| [[Dual (category theory)|Dually]], the [[coproduct]] of an empty family is an [[initial object]].
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| Nullary categorical products or coproducts may not exist in a given category; e.g. in the [[category of fields]], neither exists.
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| == In computer programming ==
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| Many programming languages, such as [[Python (programming language)|Python]], allow the direct expression of lists of numbers, and even functions that allow an arbitrary number of parameters. If such a language has a function that returns the product of all the numbers in a list, it usually works like this:
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| listprod( [2,3,5] ) --> 30
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| listprod( [2,3] ) --> 6
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| listprod( [2] ) --> 2
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| listprod( [] ) --> 1
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| This convention sometimes helps avoid having to code special cases like "if length of list is 1" or "if length of list is zero" as special cases.
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| Many programming languages do not permit the direct expression of the empty product, because they do not allow expressing lists. Multiplication is taken to be an [[infix notation|infix]] operator and therefore a binary operator. Languages implementing [[variadic function]]s are the exception. For example, the [[S-expression|fully parenthesized prefix notation]] of [[Lisp programming language|Lisp languages]] gives rise to a natural notation for [[nullary]] functions:
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| (* 2 2 2) ; evaluates to 8
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| (* 2 2) ; evaluates to 4
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| (* 2) ; evaluates to 2
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| (*) ; evaluates to 1
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| ==See also==
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| *[[Iterated binary operation]]
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| *[[Empty sum]]
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| ==References==
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| <references/>
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| == External links ==
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| * [http://planetmath.org/encyclopedia/EmptyProduct.html PlanetMath article on the empty product]
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| {{PlanetMath attribution|id=6458|title=Empty product}}
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| {{DEFAULTSORT:Empty Product}}
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| [[Category:Multiplication]]
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| [[Category:One]]
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Most individuals have this habit of doing all stuff by themselves, regardless of how critical or simple they are! These people won't let others interfere in their affairs. While this stance might work in other regions of life, it's not how to respond when you need to fix your Windows registry. There are some jobs such as removing spywares, virus plus obsolete registry entries, that are best left to specialist softwares. In this particular article I usually tell we why it happens to be critical to fix Windows registry NOW!
Google Chrome crashes on Windows 7 by the corrupted cache contents and difficulties with the stored browsing information. Delete the browsing information and obvious the contents of the cache to resolve this problem.
System tray icon makes it easy to launch the system and displays "clean" status or the amount of mistakes in the last scan. The ability to locate and remove the Invalid class keys and shell extensions is regarded as the primary advantages of the system. That is not normal function for the additional Registry Cleaners. Class keys plus shell extensions which are not working can really slow down a computer. RegCure scans to obtain invalid entries and delete them.
Chrome enables customizing itself by applying range of themes available online. If you had lately used a theme that no longer works correctly, it results inside Chrome crash on Windows 7. It is recommended to set the authentic theme.
Google Chrome crashes on Windows 7 if the registry entries are improperly modified. Missing registry keys or registry keys with wrong values can cause runtime mistakes and thereby the issue happens. We are recommended to scan the entire system registry plus review the result. Attempt the registry repair task using third-party pc tools registry mechanic software.
Windows relies heavily on this database, storing everything from the newest emails to the Internet favorites inside there. Because it's thus crucial, the computer is continually adding plus updating the files inside it. This is ok, nevertheless it can create a computer run slow, whenever the computer accidentally breaks its crucial registry files. This is a really usual issue, and actually makes a computer run slower each day. What arises is that since a computer is frequently using 100's of registry files at once, it occasionally gets confused plus make a few of them unreadable. This then makes the computer run slow, because Windows takes longer to read the files it demands.
By restoring the state of the system to an earlier date, error 1721 could not appear inside Windows 7, Vista plus XP. There is a tool called System Restore that we have to employ inside this procedure.
You are able to click here to obtain out how to accelerate Windows and grow PC perfomance. And you are able to click here to download a registry cleaner to aid you clean up registry.