Difference between revisions of "Set"

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A '''set''' is a property of [[numbers]] which can be thought of as an object but without altering their properties. However Russell's [[paradox]] states that there are no "normal" sets, insofar as they would not be contained in itself, leading to a contradiction.
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A '''set''' is a collection of objects, and sets are the subject of [[set theory]].  Two sets are the same if they contain the same elements, regardless of the order they are in. Sets are written in set notation, such as {1,2} indicating the set containing the elements 1 and 2. This is the same as the set {2,1}, as order does not matter.
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Also, repetition of elements is irrelevant, so {1,2,2} is the same as the set {1,2}.
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An important concept in set theory is [[cardinality]]. In the case of finite sets, this is simply the number of elements of the set. So {1,2} has a cardinality of 2. However, the theory is more complicated in the case of infinite sets. 
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It was thought that an informal concept of sets ([[Naive set theory]]) was sufficient, however [[Bertrand Russell's]] shows that this can lead to a contradiction, if sets are allowed to contain themselves. Modern set theory is more formal, and disallows such paradoxical sets.
  
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Revision as of 12:38, 21 December 2007

Template:Stub A set is a collection of objects, and sets are the subject of set theory. Two sets are the same if they contain the same elements, regardless of the order they are in. Sets are written in set notation, such as {1,2} indicating the set containing the elements 1 and 2. This is the same as the set {2,1}, as order does not matter.

Also, repetition of elements is irrelevant, so {1,2,2} is the same as the set {1,2}.

An important concept in set theory is cardinality. In the case of finite sets, this is simply the number of elements of the set. So {1,2} has a cardinality of 2. However, the theory is more complicated in the case of infinite sets.

It was thought that an informal concept of sets (Naive set theory) was sufficient, however Bertrand Russell's shows that this can lead to a contradiction, if sets are allowed to contain themselves. Modern set theory is more formal, and disallows such paradoxical sets.