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Mehran Sahami
Handout #4
CS103B
January 7, 2009
Introduction to Sets
Portions of this handout was originally written by Maggie Johnson, however she cannot be held responsible for the humorless
jokes now contained herein.
Remembering those halcyon days of youth back in CS106B/X, you may recall encountering the
Set abstract data type.
Now, it's time to get a little more formal (please take a moment to change
into black tie or evening gown attire…) with this concept, so that we can build on this concept
mathematically (as opposed to programmatically).
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 which has
no repetition and ignores order (i.e., an unordered collection of unique objects).
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).
Sorry if anyone reading this is from Libertyville.
Definition
: A
set
is a collection of distinct objects without repetition and without order.
The
objects in a set are called the
elements
or
members
of the set.
We can formally define the members of a set in two ways.
The first is simply to enumerate the
members, as in:
P
= {
Ramonas
,
PizzaMyHeart
}.
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, known as the
empty set
, is denoted by empty braces {} or
by the symbol
.
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
and q
0}.
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 number p/q
that satisfies the
condition to the right of the bar.
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 k >= 0 }
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Fun Facts About Sets
Here are some important facts about sets.
Learn them, know them, love them, live them…
1) Two sets are
equal
if and only if they have the same elements.
For example:
{3, 5, 1} = {5, 1, 3}
2) If A and B are sets, then A is a
subset
of B (denoted A
B) if and only if every element of
A is also an element of B.
Analogously, A is a
superset
of B (denoted A
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 Winter '08
 SAHAMI,M

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