Maggie Johnson
Handout #6
CS103B
Sets
Key topics:
Introduction and Definitions
Set Operations
Set Identities
Cartesian Product
Proofs about Sets
Set Applications
The first major abstraction that we will consider is a set.
Sets are the most basic of mathematical
structures.
They are so basic, in fact, that they are hard to define without using the word “set” or
some synonym of “set”.
The set concept allows us to talk about a collection of objects, ignoring
order and ignoring repetition.
The only relationship between the elements of a set is that they all
belong to the same set.
We can speak of the set of integers (which is infinite), the set of real
numbers (which is infinite in a different way), the set of good pizza places in Palo Alto (very finite),
or the set of good pizza places in Libertyville, IL (empty set).
Definition:
A
set
is a collection of distinct objects without repetition and without order.
The
elements of a set are referred to as its members.
We can formally define the members of a set in two ways. The first is simply to enumerate the
members, as in:
P
= {
Ramonas
,
PizzaAGoGo
}.
Note that sets are traditionally given capital letters for names.
Some important sets are given special
names such as:
R
for the real numbers
Q
for the rational numbers
Z
for the integers
N
for the natural numbers
The symbol
∈
denotes membership in a set, and
∉
denotes nonmembership in a set.
Thus,
Ramonas
∈
P
, but
JiffyLube
∉
P
, meaning that Ramona’s is a good place to buy pizza, but Jiffy
Lube is not.
A set with no members is denoted by empty braces {}.
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
∈
Z
}.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
defines the set of rational numbers. The vertical bar  is read "such that."
This definition works
because there is an understanding that
the set contains every
p
/
q
that satisfies the condition to the
right
of the bar.
Another example, the set {1, 3, 5, 7, 9, 11, ... } can be defined as:
{ n
∈
Z

n = 2k+1
for some integer k >= 0 }
Some other important facts about sets:
1) Two sets are equal
if and only if they have the same elements:
{1, 3, 5, 5, 3, 1, 1 } = {5, 1, 3}
2) If A and B are sets, then A is a
subset
of B if and only if, every element of A is also an
element of B:
A = {1,2,5,7} B = {1, 5}
C = {1, 5} B is a subset of A; B is a subset of C.
Another way of defining set equality is to say A = B if and only if A is a subset of B and B is
a subset of A.
Note that if you need to prove set equality, you have to prove two
things.
3) If A is a subset of B but A does not equal B, then A is called a
proper subset
.
In the
example above, B is a proper subset of A, but B is not a proper subset of C.
4)
Venn diagrams
are often used to graphically represent sets.
In Venn diagrams, the
Universal Set (U) contains all the objects under consideration.
This is represented by a
rectangle.
Inside the rectangle are circles or other geometrical figures used to represent
sets.
Venn diagrams are often used to illustrate the relationship between sets.
For example,
the first diagram below shows that A is a subset of B.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 SAHAMI,M
 Sets, John Venn

Click to edit the document details