Handout #14
CS103
April 14, 2010
Robert Plummer
Introduction to Sets
Sets are the most basic of mathematical structures. It is possible to build all of mathematics on just
the concept set membership and a small number of axioms.
It is important to know that
mathematics has such a foundation, but "axiomatic set theory" is too tedious to develop for our
purposes.
Our approach will be a more intuitive one, where we assume that we know and understand what
objects are, what collections of objects are, and what it means for an object to belong to a collection.
This approach is called "naive set theory", and in spite of that name, it provides us with all the
formality and rigor we will need.
We start with the following definitions:
Definitions
A
set
is an unordered collection of distinct objects.
The objects in a set are called the
elements
or
members
of the set.
If a set has exactly
n
members for some nonnegative integer
n
, then we say that the set is
finite
.
The number of members of a set is called its
size
, or
cardinality
.
The set with no members is called the
empty set
and is denoted by the symbol
∅
or by empty
braces {}.
A set that is not finite is said to be
infinite
.
The size of an infinite set is greater than any integer.
This fits our intuitive notion of a set.
For example, we could consider the set of all students
registered for CS103.
Each student is distinct from all the others, and no ordering of the students is
implied.
There are exactly 151 students enrolled at the moment, so the cardinality of the set of
CS103 students is 151, and the set is finite.
None of the TAs are enrolled, so the set of CS103
students who are also TAs is the empty set.
By contrast, the set of all nonnegative integers is
infinite.
We can formally define the members of a set in two ways.
The first is simply to enumerate the
members by listing them within brackets. For example:
CS103 TAs
= {
Karl, Mridul, Steve, Tyler
}
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The second way to define a set is to specify some condition for membership. This is referred to as
set
builder notation
.
For example:
Q
= { p/q  p, q are integers and q
≠
0}.
defines the set of rational numbers. The slash / means division and the vertical bar  is read "such
that."
This definition means that the set contains every number p/q
that satisfies the condition to the
right of the bar.
Note that 2/4, 3/6, 4/8, ... are simply alternate ways of specifying the unique
member of the set that we normally write as 1/2.
Note also that Q is infinite.
As another example, the set of positive odd integers {1, 3, 5, 7, 9, 11, ... } can be defined as:
{ n  n = 2k+1 for integer some k
≥
0 }
Sets are traditionally given capital letters for names.
Some important sets are given special names:
R
for the real numbers
Q
for the rational numbers
Z
for the integers
N
for the natural numbers {1, 2, 3, ...}
The symbol
∈
denotes membership in a set, and
∉
denotes nonmembership in a set.
Thus,
2.51
∈
Q, but 2.51
∉
N.
The symbol
∈
looks somewhat like the Greek letter epsilon, but it is
really a special math symbol.
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 Fall '09
 Set Theory, Sets, John Venn

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