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