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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>
| | Making the computer run swiftly is actually very easy. Most computers run slow because they are jammed up with junk files, which Windows has to look through each time it wants to obtain something. Imagine needing to discover a book in a library, however, all the library books are inside a big big pile. That's what it's like for a computer to obtain anything, whenever a system is full of junk files.<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents plus issues with all the stored browsing information. Delete the browsing data plus clear the contents of the cache to solve this issue.<br><br>Although this problem affects millions of computer consumers throughout the world, there is an effortless means to fix it. We see, there's 1 reason for a slow loading computer, and that's because a PC cannot read the files it requirements to run. In a nutshell, this simply means that whenever we do anything on Windows, it must read up on how to do it. It's traditionally a very 'dumb' program, that has to have files to tell it to do everything.<br><br>Analysis a files plus clean it up regularly. Destroy all of the unwanted plus unused files because they just jam your computer program. It will surely improve the speed of your computer plus be thoughtful which the computer do not afflicted by a virus. Remember constantly to update your antivirus software every time. If you do not utilize the computer rather usually, you are able to take a free antivirus.<br><br>Another thing you need to check is whether the [http://bestregistrycleanerfix.com/system-mechanic system mechanic] program you are considering has the ability to identify files plus programs that are wise. One of the registry cleaner programs you could try is RegCure. It is helpful for speeding up plus cleaning up issues on the computer.<br><br>Let's begin with the bad sides initially. The initial cost of the product is truly inexpensive. However, it only comes with one year of updates. After which we must subscribe to monthly updates. The benefit of that is that ideal optimizer has enough income and resources to analysis mistakes. This method, you may be ensured of safe fixes.<br><br>The reason why this really is significant, is because various of the 'dumb' registry products really delete these files without even recognizing. They simply browse through the registry plus try plus find the many problems possible. They then delete any files they see fit, plus because they are 'dumb', they don't actually care. This means that when they delete some of these vital system files, they are really going to cause a LOT more damage than wise.<br><br>If you want to have a computer with quick running speed, you'd better install a wise registry cleaner to clean the useless files for you. As long as you take care of your computer, it usually keep in advantageous condition. |
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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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Making the computer run swiftly is actually very easy. Most computers run slow because they are jammed up with junk files, which Windows has to look through each time it wants to obtain something. Imagine needing to discover a book in a library, however, all the library books are inside a big big pile. That's what it's like for a computer to obtain anything, whenever a system is full of junk files.
Google Chrome crashes on Windows 7 by the corrupted cache contents plus issues with all the stored browsing information. Delete the browsing data plus clear the contents of the cache to solve this issue.
Although this problem affects millions of computer consumers throughout the world, there is an effortless means to fix it. We see, there's 1 reason for a slow loading computer, and that's because a PC cannot read the files it requirements to run. In a nutshell, this simply means that whenever we do anything on Windows, it must read up on how to do it. It's traditionally a very 'dumb' program, that has to have files to tell it to do everything.
Analysis a files plus clean it up regularly. Destroy all of the unwanted plus unused files because they just jam your computer program. It will surely improve the speed of your computer plus be thoughtful which the computer do not afflicted by a virus. Remember constantly to update your antivirus software every time. If you do not utilize the computer rather usually, you are able to take a free antivirus.
Another thing you need to check is whether the system mechanic program you are considering has the ability to identify files plus programs that are wise. One of the registry cleaner programs you could try is RegCure. It is helpful for speeding up plus cleaning up issues on the computer.
Let's begin with the bad sides initially. The initial cost of the product is truly inexpensive. However, it only comes with one year of updates. After which we must subscribe to monthly updates. The benefit of that is that ideal optimizer has enough income and resources to analysis mistakes. This method, you may be ensured of safe fixes.
The reason why this really is significant, is because various of the 'dumb' registry products really delete these files without even recognizing. They simply browse through the registry plus try plus find the many problems possible. They then delete any files they see fit, plus because they are 'dumb', they don't actually care. This means that when they delete some of these vital system files, they are really going to cause a LOT more damage than wise.
If you want to have a computer with quick running speed, you'd better install a wise registry cleaner to clean the useless files for you. As long as you take care of your computer, it usually keep in advantageous condition.