added a bunch of math content based on alaska notes. added more ethernet content

1 files changed, 40 insertions, 5 deletions

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.