From 397cefef7929e6cd49959db1d891f5b0654ebd05 Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Tue, 10 Jun 2008 10:48:24 -0400 Subject: added a bunch of math content based on alaska notes. added more ethernet content --- math/sets | 45 ++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 40 insertions(+), 5 deletions(-) (limited to 'math/sets') diff --git a/math/sets b/math/sets index 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. -- cgit v1.2.3