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@@ -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. |