A Probability Course for the Actuaries A Preparation for Exam P/1

26 BASIC OPERATIONS ON SETS
Example 2.13
A total of 35 programmers interviewed for a job; 25 knew FORTRAN, 28
knew PASCAL, and 2 knew neither languages. How many knew both languages
Solution.
Let F be the group of programmers that knew FORTRAN, P those who
knew PASCAL. Then F ∩ P is the group of programmers who knew both
languages. By the Inclusion-Exclusion Principle we have n(F ∪ P ) = n(F ) +
n(P ) − n(F ∩ P ). That is, 33 = 25 + 28 − n(F ∩ P ). Solving for n(F ∩ P ) we
find n(F ∩ P ) = 20
Cartesian Product
The notation (a, b) is known as an ordered pair of elements and is defined
by (a, b) = {{a}, {a, b}}.
The Cartesian product of two sets A and B is the set
A × B = {(a, b)|a ∈ A, b ∈ B}.
The idea can be extended to products of any number of sets. Given n sets
A 1 , A 2 , · · · , A n the Cartesian product of these sets is the set
A 1 × A 2 × · · · × A n = {(a 1 , a 2 , · · · , a n ) : a 1 ∈ A 1 , a 2 ∈ A 2 , · · · , a n ∈ A n }
Example 2.14
Consider the experiment of tossing a fair coin n times. Represent the sample
space as a Cartesian product.
Solution.
If S is the sample space then S = S 1 × S 2 × · · · × S n where S i , 1 ≤ i ≤ n is
the set consisting of the two outcomes H=head and T = tail
The following theorem is a tool for finding the cardinality of the Cartesian
product of two finite sets.
Theorem 2.4
Given two finite sets A and B. Then
n(A × B) = n(A) · n(B).