2
Sets
Before proceeding, we will give a few definitions concerning sets:
A set
is an unordered collection of distinct objects, which we call the elements of the set.
The set of no elements is called the empty set
.
If A is a finite set, A denotes the number of elements in A, which is called the cardinality
of A.
The union
of sets A and B, denoted A
∪
B, is the set of all elements in A or B.
The intersection
of sets A and B, denoted A
∩
B, is the set of all elements in both A and B.
Generalized Sum and Product Rules
The Sum Rule
:
If a task can be accomplished by choosing one of the alternatives from the sets
S
1
, S
2
, .
.., S
m
, and these sets are pairwise disjoint (i.e., S
i
∩
S
j
= Ø for all i
≠
j), and n
i
is the
number of elements in S
i
, then the number of ways to accomplish the task is n
1
+ n
2
+ .
.. + n
m
.
Using the notation of set theory, we would write S
1
∪
S
2
∪
...
∪
S
m
 = S
1
 + S
2
 + .
.. + S
m

(where the sets are disjoint).
The Product Rule
:
If E
1
, E
2
, .
.., E
m
is a sequence of events such that E
1
can occur in n
1
ways and
if E
1
, E
2
, .
.., E
k1
have occurred, then E
k
can occur in n
k
ways, then there are n
1
n
2
...n
m
ways in
which the entire sequence of events can occur.
Suppose you are either going to go to an Italian restaurant that
serves 15 entrées or to a French restaurant that serves 10 entrées.