Difference between revisions of "Class"

(New page: In set theory, a class refers to a set made up of other sets. For example, if <math>A=\{a,b,c\}</math> is a set, then <math>\mathscr{A}=\{\{a\},\{b\},\{b,c\}\}</math> is a class ma...)
 
 
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In [[set theory]], a class refers to a [[set]] made up of other sets. For example, if <math>A=\{a,b,c\}</math> is a set, then <math>\mathscr{A}=\{\{a\},\{b\},\{b,c\}\}</math> is a class made up of some of the subsets of <math>A</math>.
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In [[set theory]], a class is essentially a set which we do not call a set for logical or semantic reasons. For example, if <math>A=\{a,b,c\}</math> is a set, then <math>B=\{\{a\},\{b\},\{b,c\}\}</math> is a class consisting of some subsets of <math>A</math>. In this example though, <math>B</math> can also be called a set.
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To understand why one would make such a distinction, consider Russell's Paradox:
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"Define <math>T</math> to be the set of all sets which do not contain themselves. Is it true or not that <math>T\in T</math>?" If <math>T\in T</math>, then <math>T</math> must not contain itself; that is, <math>T\not\in T</math>. If <math>T\not\in T</math>, then it must be because <math>T \in T</math>. Either way there is a contradiction. One resolution to Russell's paradox changes the language every so slightly: "Define <math>T</math> to be the class of all sets which do not contain themselves. Is it true or not that <math>T\in T</math>?" Indeed, <math>T\not\in T</math> for <math>T</math> is not a set.
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Compare Russell's Paradox to the Barber of Seville problem: "The barber of Seville shaves exactly those men who do not shave themselves. How can this be?" Naturally, the barber is a woman.
  
 
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[[Category:Mathematics]]
 
[[Category:Mathematics]]

Latest revision as of 14:28, 11 August 2008

In set theory, a class is essentially a set which we do not call a set for logical or semantic reasons. For example, if $A=\{a,b,c\}$ is a set, then $B=\{\{a\},\{b\},\{b,c\}\}$ is a class consisting of some subsets of $A$. In this example though, $B$ can also be called a set.

To understand why one would make such a distinction, consider Russell's Paradox: "Define $T$ to be the set of all sets which do not contain themselves. Is it true or not that $T\in T$?" If $T\in T$, then $T$ must not contain itself; that is, $T\not\in T$. If $T\not\in T$, then it must be because $T \in T$. Either way there is a contradiction. One resolution to Russell's paradox changes the language every so slightly: "Define $T$ to be the class of all sets which do not contain themselves. Is it true or not that $T\in T$?" Indeed, $T\not\in T$ for $T$ is not a set.

Compare Russell's Paradox to the Barber of Seville problem: "The barber of Seville shaves exactly those men who do not shave themselves. How can this be?" Naturally, the barber is a woman.

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