CHAPTER 4
Applications of Methods of Proof
1. Set Operations
1.1. Set Operations.
The set-theoretic operations, intersection, union, and
complementation, deﬁned in
Chapter 1.1 Introduction to Sets
are analogous to the
operations
∧
,
∨
, and
¬
, respectively, that were deﬁned for propositions. Indeed, each
set operation was deﬁned in terms of the corresponding operator from logic. We will
discuss these operations in some detail in this section and learn methods to prove
some of their basic properties.
Recall that in any discussion about sets and set operations there must be a set,
called a
universal set
, that contains all others sets to be considered. This term is a
bit of a misnomer: logic prohibits the existence of a “set of all sets,” so that there
is no one set that is “universal” in this sense. Thus the choice of a universal set will
depend on the problem at hand, but even then it will in no way be unique. As a rule
we usually choose one that is minimal to suit our needs. For example, if a discussion
involves the sets
{
1
,
2
,
3
,
4
}
and
{
2
,
4
,
6
,
8
,
10
}
, we could consider our universe to be
the set of natural numbers or the set of integers. On the other hand, we might be
able to restrict it to the set of numbers
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
.
We now restate the operations of set theory using the formal language of logic.
1.2. Equality and Containment.
Definition
1.2.1
.
Sets
A
and
B
are
equal
, denoted
A
=
B
, if
∀
x
[
x
∈
A
↔
x
∈
B
]
Note: This is equivalent to
∀
x
[(
x
∈
A
→
x
∈
B
)
∧
(
x
∈
B
→
x
∈
A
)]
.
Definition
1.2.2
.
Set
A
is contained in set
B
(or
A
is a subset of
B
), denoted
A
⊆
B
,if
∀
x
[
x
∈
A
→
x
∈
B
]
.
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