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| author | User <bnewbold@daemon.robocracy.org> | 2009-10-13 02:52:09 +0000 |
|---|---|---|
| committer | User <bnewbold@daemon.robocracy.org> | 2009-10-13 02:52:09 +0000 |
| commit | f61026119df4700f69eb73e95620bc5928ca0fcb (patch) | |
| tree | f17127cff9fec40f4207d9fa449b9692644ce6db /math/sets | |
| parent | 9d431740a3e6a7caa09a57504856b5d1a4710a14 (diff) | |
| download | knowledge-f61026119df4700f69eb73e95620bc5928ca0fcb.tar.gz knowledge-f61026119df4700f69eb73e95620bc5928ca0fcb.zip | |
Grand rename for gitit transfer
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| -rw-r--r-- | math/sets | 47 |
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diff --git a/math/sets b/math/sets deleted file mode 100644 index 42eb831..0000000 --- a/math/sets +++ /dev/null @@ -1,47 +0,0 @@ -==================== -Sets -==================== - -.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ - -Basics -============= -If every element :latex:`$a \in A$` is also :latex:`$a \in B$`, then we call -A a *subset* of B and write :latex:`$A \subset B$`. If there are elements of B -which are not elements of A, then we call A a *proper subset* of B. - -If :latex:`$A \supset B$` and :latex:`$B \supset A$` we write :latex:`$A = B$`; -otherwise :latex:`$A \neq B$`. - -The null or empty set, which has no elements, is a subset of all others. - -A relation on a space of sets S is something that can be definted as either -true or false (holding or not holding) for any binary pair in S. - -Binary Operators -================== -Binary operators defined on a set apply to any two elements of that set; order -may or may not be important. A set is *closed* with regards to a binary -operator if it contains the result of the binary operator. A set is *uniquely -defined* with regards to a binary operator if the result of the operator on two -elements of the set is unique from the results from all other pairs of -elements. - -Some equivalence relations are -:latex:`$\identity$` (NOTE: = with three lines) (*identity*); -:latex:`$\congruence$` (NOTE: = with tilde on top) (*congruence*; eg of -geometric figures); and -:latex:`$~$` (NOTE: tilde) (*similarity*; eg of geometric figures). - -Some properties of equivalence relations are - -*reflexive* - if :latex:`$a=a$` is true for all a -*symetric* - if :latex:`$a=b$` implies :latex:`$b=a$` -*transitive* - if :latex:`$a=b$` and :latex:`$b=c$` implies :latex:`$a=c$` - -.. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976 - -.. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve. |