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author committer bryan newbold 2008-06-10 10:48:24 -0400 bryan newbold 2008-06-10 10:48:24 -0400 397cefef7929e6cd49959db1d891f5b0654ebd05 (patch) ec197475d62c9149288863f89c97e26131959815 /math/sets 80b4628e5a94dfedd2c6ee91bdb608c531a78598 (diff) knowledge-397cefef7929e6cd49959db1d891f5b0654ebd05.tar.gzknowledge-397cefef7929e6cd49959db1d891f5b0654ebd05.zip
 diff --git a/math/sets b/math/setsindex 6d75a55..42eb831 100644--- a/math/sets+++ b/math/sets@@ -2,11 +2,46 @@ Sets ==================== -.. warning:: Under progress!+.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ -.. note:: Most of the definitions and notation in the section are based on [rudin]_+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:$A \neq B$.+If :latex:$A \supset B$ and :latex:$B \supset A$ we write :latex:$A = B$;+otherwise :latex:$A \neq B$. -.. [rudin] Principles of Mathematical Analysis (3rd ed):title:, by Walter Rudin.- McGraw-Hill, 1976+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.