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*References: Most of the definitions and notation in the section are based on
[^rudin] or [^meserve]*
If every element $a \in A$ is also $a \in B$, then we call
A a *subset* of B and write $A \subset B$. If there are elements of B
which are not elements of A, then we call A a *proper subset* of B.
If $A \supset B$ and $B \supset A$ we write $A = B$;
otherwise $A \neq B$.
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
$\identity$ (NOTE: = with three lines) (*identity*);
$\congruence$ (NOTE: = with tilde on top) (*congruence*; eg of
geometric figures); and
$~$ (NOTE: tilde) (*similarity*; eg of geometric figures).
Some properties of equivalence relations are
if $a=a$ is true for all a
if $a=b$ implies $b=a$
if $a=b$ and $b=c$ implies $a=c$
[^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976
[^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.