Sets will usually be denoted by capital letters,
,
,
..., ,
,
..., ,
,
... . Lower
case letters will usually denote elements of sets. The symbol
denotes
set membership; while the symbol
denotes ``not a member of.'' The
English sentences ``
is an integer'' and ``
is not an
integer'' are written in symbols as ``
'' and ``
''. Both statements are true. Whereas the statement ``
'' is not true. In the most common form of logic, the one we use, most
statements^{1} are either
true or not true. This does not imply that we can always
determine whether a given statement is true or not. Among other common
symbols we use are
union,
intersection,
there
exists,
there exists a unique,
for all. We will often
describe sets by expressions like

For example,

(the use of the semicolon ; in the above expression is equivalent to the words such that - that will often be abbreviated s.t.) and

The same set may be described in more than one way:

The formal definitions of , set theoretic

and

Technically, the definition of and the first of our two definitions of are not correct since they do not specify from which set is chosen. All our operations must for technical foundational issues be restricted to some universal set. In practical terms, this issue seldom causes any problems for the topics that we will study. Two more symbols that are commonly used involve set