Difference between revisions of "Set"
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| − | A '''set''' is a collection of | + | 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}. | |
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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 [[Russell's Paradox]] 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 08:48, September 12, 2008
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 Russell's Paradox 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.