==================== 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.