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In [[set theory]], an '''infinite set''' is a [[Set (mathematics)|set]] that is not a [[finite set]]. Infinite sets may be [[countable set|countable]] or [[uncountable set|uncountable]]. Some examples are:
* the set of all [[integer]]s, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
* the set of all [[real number]]s is an [[uncountable set|uncountably infinite set]].
 
==Properties==
The set of [[natural numbers]] (whose existence is postulated by the [[axiom of infinity]]) is infinite. It is the only set that is directly required by the [[axiom]]s to be infinite. The existence of any other infinite set can be proved in [[Zermelo–Fraenkel set theory]] (ZFC) only by showing that it follows from the existence of the natural numbers.
 
A set is infinite if and only if for every natural number the set has a [[subset]] whose [[cardinality]] is that natural number.
 
If the [[axiom of choice]] holds, then a set is infinite if and only if it includes a countable infinite subset.
 
If a [[set of sets]] is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set is infinite. Any [[subset|superset]] of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The [[Cartesian product]] of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite.
 
If an infinite set is a [[well-ordered set]], then it must have a nonempty subset that has no greatest element.
 
In ZF, a set is infinite if and only if the [[powerset]] of its powerset is a [[Dedekind-infinite set]], having a proper subset [[equinumerous]] to itself. If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.
 
If an infinite set is a [[well-orderable set]], then it has many well-orderings which are non-isomorphic.
 
==History==
The first known{{citation needed|date=September 2011}} occurrence of explicitly infinite sets is in Galileo's last book [[Two New Sciences]] written while he was under house arrest by the [[Inquisition]].<ref>Drake, Stillman, translator (1974). Two New Sciences, University of Wisconsin Press, 1974. ISBN 0-299-06404-2. A new translation including sections on centers of gravity and the force of percussion.</ref>
 
Galileo argues that the set of squares <math>\mathbb{S} = \{1,4,9,16,25,\ldots\}</math> is the same size as <math>\mathbb{N} = \{1,2,3,4,5,\ldots\}</math> because there is a one-to-one correspondence:
 
:<math>1 \leftrightarrow 1, 2 \leftrightarrow 4, 3 \leftrightarrow 9, 4 \leftrightarrow 16, 5 \leftrightarrow 25, \ldots</math>
 
And yet, as he says, <math>\mathbb{S}</math> is a proper subset of <math>\mathbb{N}</math> and <math>\mathbb{S}</math> even gets less dense as the numbers get larger.
 
==See also==
* [[Dedekind-infinite set]]
* [[Aleph number]]
* [[Two New Sciences|Galileo's ''Two New Sciences'']]
* [[Infinity]]
 
==References==
{{Reflist}}
 
== External links ==
*{{MathWorld |title=Infinite Set |id=InfiniteSet }}
 
{{Set theory}}
 
[[Category:Basic concepts in infinite set theory| ]]
[[Category:Cardinal numbers]]

Latest revision as of 19:06, 12 January 2015

Have we been thinking "how do I speed up my computer" lately? Well chances are in the event you are reading this article; then you are experiencing one of countless computer issues that thousands of individuals discover that they face on a regular basis.

We all learn which the registry is the critical component of the Windows running program because it stores all information regarding the Dll files, programs found on the computer plus system settings. However, because days by, it is unavoidable which we might encounter registry matter due to a huge amount of invalid, useless plus unwelcome entries.

The 'registry' is a central database which stores info, settings and options for the computer. It's actually the most normal reason why XP runs slow plus should you fix this issue, you could create the computer run a lot faster. The problem is that the 'registry' stores a lot of settings and details about the PC... plus considering Windows must use numerous of these settings, any corrupted or damaged ones might straight affect the speed of the system.

It is normal that the imm32.dll error is caused as a result of a mis-deletion activity. If you cannot discover the imm32.dll anywhere on the computer, there is no question that it should be mis-deleted when uninstalling programs or other unneeded files. Hence, you are able to directly cope it from alternative programs or download it from a safe web and then put it on a computer.

These are the results that the tuneup utilities 2014 found: 622 incorrect registry entries, 45,810 junk files, 15,643 unprotected privacy files, 8,462 bad Active X products that have been not blocked, 16 performance features which were not optimized, plus 4 changes which the computer needed.

The key reason why I couldn't make my PC run faster was the system registry and it being fragmented. So software to defragment or clean the registry are required. Such software are called registry products. Like all other software, there are paid ones and free ones with their blessings and disadvantages. To choose between the two is the user's choice.

As the hub center of the computer, the important settings are stored the registry. Registry is structured as keys plus every key relates to a system. The system reads the keys plus utilizes the information to launch plus run programs. However, the big problem is that there are too many unwanted settings, useless info occuping the useful space. It makes the system run slowly and big amounts of settings become unreadable.

Thus, the best thing to do when a computer runs slow is to purchase an authentic and legal registry repair tool which would aid you eliminate all difficulties connected to registry plus help we enjoy a smooth running computer.

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