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Chapter 3 – Set Theory

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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.

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Example 3.1: page 128.

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Example 3.2: page 128.

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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.

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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|.

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Example 3.3: page 129.

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Example 3.4: page 129.

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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).

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Example page 130.

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Example 3.5

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**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.

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Proof of Theorem 3.1

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Example 3.6: page 131.

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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.)

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Theorem 3.2: For any universe U, let AU. Then A, and if A, then A.

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Example 3.7: page 132.

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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.

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Example 3.8: page 132.

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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.

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Example 3.9: page 133.

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Example 3.10

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Example 3.11: page 135. (Note: )

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**Example 3.13: page 136. (Pascal’s triangle)**

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**3.2 Set Operations and the Laws of Set Theory**

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**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 | (xA xB) xAB} = {x | xAB xAB}. 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.

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Example 3.14: page 140.

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Definition 3.6: Let S, T U. The sets S and T are called disjoint, or mutually disjoint, when S T = .

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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.

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Definition 3.7: For a set A U, the complement of A denote U – A, or , is given by {x | xU xA}.

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Example 3.15: page 141.

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Definition 3.8: A, B U, the (relative) complement of A in B, denoted B – A, is given by {x | xB xA}.

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Example 3.16: page 141.

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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’

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**The Laws of Set Theory: page 142~143.**

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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.

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**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.

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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.)

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Membership table: We observe that for sets A, B U, an element 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 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.

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**(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.

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Example 3.18: page 148.

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Example 3.19: page 148.

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Example 3.20

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Example 3.21

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Example 3.22

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**3.3 Counting and Venn Diagrams**

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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).

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**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|.

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**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.

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3.4 A Word on Probability

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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 =.

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Example 3.26: page 154.

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Example 3.27: page 155.

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Example 3.29: page 155~156.

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