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+.. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_
+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
+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
+ if :latex:`$a=a$` is true for all a
+ if :latex:`$a=b$` implies :latex:`$b=a$`
+ 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.