Why Read This Book? - Index of

78 Chapter 3 Sets and Their Properties
Then
1. ∪F2A ⊆∪F1A 2. ∩F2A ⊇∩F1A The next theorem involves two families of sets, both indexed by A, where
corresponding sets in the two families are related by subset inclusion. To understand
what it says, think of Bα as the set of all female PU graduates who passed
mathematics course number α.
EXERCISE 3.3.12 Suppose F1 ={Aα}α∈A and F2 ={Bα}α∈A are two families
of sets with the property that Bα ⊆ Aα for every α ∈ A. Then
(a) �
α∈A Bα ⊆ �
α∈A Aα
(b) �
α∈A Bα ⊆ �
α∈A Aα
EXERCISE 3.3.13 Suppose F ={A} is a family of sets, and suppose C is a set
for which A ⊆ C for every A ∈ F. Show that ∪F A ⊆ C. 5
EXERCISE 3.3.14 Suppose F ={A} is a family of sets, and suppose D is a set
for which D ⊆ A for every A ∈ F. Show that D ⊆∩F A. 6
3.4 The Principle of Mathematical Induction
In the world of mathematics, the well-ordering principle (WOP) is often taken
as an axiom. In this section, we derive a theorem based on the WOP called the
Principle of Mathematical Induction (PMI). To write a proof by induction, the
imagery is that we have an infinite row of dominoes that we must knock down.
First, we show figuratively that we can knock the first domino down. Then we
show that if the nth domino falls, then so does the (n + 1)st. This very important
proof technique is useful when the theorem you’re trying to prove has a form like
any of the following examples.
Example 3.4.1 For any positive integer n,
1 + 2 + 3 +···+n =
n�
k =
k=1
n(n + 1)
2
(3.34)
5 Think of C as the set of all PU graduates who ever enrolled in a mathematics class.
6 Think of D as the set of all PU mathematics majors (whom we will assume would have taken every
mathematics course) who graduated with a 4.00 GPA.
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