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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 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.