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Chapter 3 – Set Theory
3.1 Sets and Subsets
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
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
A set is a well-defined collection of objects. These
objects are called elements and are said to be
members of the set.
For a set A, we write x  A if x is an element of A; y
 A indicated that y is not a member of A.
A set can be designated by listing its elements
within set braces, e.g., A = {1, 2, 3, 4, 5}.
Another standard notation for this set provides us
with A = {x | x is an integer and 1  x  5}. Here
the vertical line | within the set braces is read “such
that”. The symbols {x |…} are read “the set of all x
such …”. The properties following | help us
determine the elements of the set that is being
described.
Example 3.1: page 128.
Example 3.2: page 128.
Example 3.2 page 128

From the above example, A and B are
examples of finite sets, where C is an
infinite set. For any finite set A, |A|
denotes the number of elements in A
and is referred to as the cardinality, or
size, of A, e.g., |A| = 9, |B| = 4.
Definition 3.1:

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
If C, D are sets from a universe U, we say that C is
a subset of D and write C  D, or D  C, if every
element of C is an element of D. If, in addition, D
contains an element that is not in C, then C is called
a proper subset of D, and this is denoted by C  D
or D  C.
Note: 1) For all sets C, D from a universe U, if C 
D, then x [x  C  x  D],
and if x [x  C  x  D], then C  D.
That is, C  D  x [x  C  x  D].
2) For all subsets C, D of U, C  D  C  D.
3) When C, D are finite, CD  |C||D|, and
CD  |C|<|D|.
Example 3.3: page 129.
Example 3.4: page 129.
Definition 3.2:
For a given universe U, the sets C and
D (taken from U) are said to be equal,
and we write C = D, when C  D and
D  C.
 Note: Some notions from logic: page
130 (line 4 from top).

Example page 130.
Example 3.5
Theorem 3.1:
Let A, B, C  U,
 a) If AB and BC, then AC.
 b) If AB and BC, then AC.
 c) If AB and BC, then AC.
 d) If AB and BC, then AC.

Proof of Theorem 3.1
Example 3.6: page 131.
Definition 3.3:

The null set, or empty set, is the
(unique) set containing no elements. It
is denoted by  or { }. (Note that ||=0
but {0}. Also, {} because {} is
a set with one element, namely, the
null set.)
Theorem 3.2:

For any universe U, let AU. Then A, and if
A, then A.
Example 3.7: page 132.
Definition 3.4:

If A is a set from universe U, the power
set of A, denoted (A), is the
collection (or set) of all subsets of A.
Example 3.8: page 132.
Lemma:

For any finite set A with |A| = n  0, we
find that A was 2n subsets and that
|(A)| = 2n. For any 0  k  n, there
are subsets of size k. Counting the
subsets of A according to the number,
k, of elements in a subset, we have the
combinatorial identity , for n  0.
n n
n n n
n
         ...         2 n
 0  1  2
n
k 0  k 
Example 3.9: page 133.
Example 3.10
Example 3.11: page 135.
(Note: )
Example 3.13: page
136. (Pascal’s triangle)
3.2 Set Operations and
the Laws of Set Theory
Definition 3.5:
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
For A, B  U we define the followings:
A  B (the union of A and B) = {x | x  A  x  B }.
A  B (the intersection of A and B) = {x | x  A  x
 B }.
A  B (the symmetric difference of A and B) = {x |
(xA  xB)  xAB} = {x | xAB  xAB}.
Note: If A, B  U, then A  B, A  B, A  B  U.
Consequently, , , and  are closed binary
operations on (A), and we may also say that (A)
is closed under these (binary) operations.
Example 3.14: page
140.
Definition 3.6:

Let S, T  U. The sets S and T are
called disjoint, or mutually disjoint,
when S  T = .
Theorem 3.3:


If S, T  U, then S and T are disjoint if and
only if S  T = S  T.
proof) proof by contradiction.
Definition 3.7:

For a set A  U, the complement of A
denote U – A, or , is given by {x | xU
 xA}.
Example 3.15: page
141.
Definition 3.8:

A, B  U, the (relative) complement of
A in B, denoted B – A, is given by {x |
xB  xA}.
Example 3.16: page
141.
Theorem 3.4:
For any universe U and any sets A, B
 U, the following statements are
equivalent:
 a) A  B
b) A  B = B
 c) A  B = A
d) B’  A’

The Laws of Set
Theory: page 142~143.
Definition 3.9:

Let s be a (general) statement dealing with
the equality of two set expressions. Each
such expression may involve one or more
occurrences of sets (such as A, , B, , etc.),
one or more occurrences of  and U, and
only the set operation symbols  and .
The dual of s, denoted sd, is obtained from s
by replacing (1) each occurrence of  and
U (in s) by U and , respectively; and (2)
each occurrence of  and  (in s) by  and
, respectively.
Theorem 3.5: The
Principle of Duality.

Let s denote a theorem dealing with
the equality of two set expressions
(involving only the set operations 
and  as described in Definition 3.9).
Then sd, the dual of s, is also a
theorem.
Venn diagram

Venn diagram is constructed as
follows: U is depicted as the interior of
a rectangle, while subsets of U are
represented by the interiors of circles
and other closed curves. (See Fig 3.5
and 3.6, page 145.)
Membership table:
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We observe that for sets A, B  U, an
element xU satisfies exactly one of the
following four situations:
a) xA, xB b) xA, xB
c) xA, xB d) xA, xB.
When x is an element of a given set, we
write a 1 in the column representing that set
in the membership table; when x is not in
the set, we enter a 0. See Table 3.2 and 3.3,
page 147.
(1) A Venn diagram is simply a
graphical representation of a
membership table.
 (2) The use of Venn diagrams and/or
membership tables may be appealing,
especially to the reader who presently
does not appreciate writing proofs.

Example 3.18: page
148.
Example 3.19: page
148.
Example 3.20
Example 3.21
Example 3.22
3.3
Counting and
Venn Diagrams

Fig 3.8 (page 152) demonstrates and ,
so by the rule of sum, |A| + || = |U| or ||
= |U| － |A|. If the sets A, B have
empty intersection, Fig 3.9 shows |A 
B| = |A| + |B|; otherwise, |A  B| = |A|
+ |B| － | A  B| (Fig 3.10).
Lemma:
If A and B are finite sets, then
 |A  B| = |A| + |B| － | A  B|.
 Consequently, finite sets A and B are
(mutually) disjoint if and only if |A  B|
= |A| + |B|.
 In addition, when U is finite, from
DeMorgan’s Law we have
 || = || = |U|－|A  B| = |U|－|A|－
|B|+|A  B|.

Lemma:
If A, B, C are finite sets, then
 .
 From the formula for |A  B  C| and
DeMorgan’s Law, we find that if the
universe U is finite, then
 Example 3.25: page 153~154.

3.4
A Word on
Probability
Lemma:
Under the assumption of equal
likelihood, let Φ be a sample space for
an experiment Ε. Any subset A of Φ is
called an event. Each element of Φ is
called an elementary event, so if |Φ| =
n and a  Φ, A  Φ, then
 Pr(a) = The probability that a occurs =,
and
 Pr(A) = The probability that A occurs =.

Example 3.26: page
154.
Example 3.27: page
155.
Example 3.29: page
155~156.