Chapter 3 – Set Theory

 

3.1 Sets and Subsets 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: 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, CD |C||D|, and

CD |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 AB and BC, then AC. b) If AB and BC, then AC. c) If AB and BC, then AC. d) If AB and BC, then AC.

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 AU. 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.

nn

k k

n

n

nnnn2...

210 0

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: 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 xB) xAB} = {x | xAB xAB}. Note: If A, B U, then A B, A B, A B U. C

onsequently, , , 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 | xU xA}.

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 | xB xA}.

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: We observe that for sets A, B U, an elem

ent xU satisfies exactly one of the following four situations:

a) xA, xB b) xA, xB c) xA, xB d) xA, xB. When x is an element of a given set, we writ

e 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 (m

utually) 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 likelihoo

d, 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.