Chapter 3 Set Theory

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Chapter 3Set Theory

• Yen-Liang Chen
• Dept of Information Management
• National Central University

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3.1 Sets and subsets

• Definitions
• Element and set , Ex 3.1
• Finite set and infinite set, cardinality ? A ?,
  Ex 3.2
• C?D a subset, C?D a proper subset
• CD, two sets are equal
• Neither order nor repetition is relevant for a
  general set
• null set, , ?

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Subset relations

• A?B
• ? ?x x?A?x?B
• A B
• ? ??x x?A?x?B
• ? ?x ? x?A?x?B
• ? ?x ? (?x?A)?(x?B)
• ? ?x x?A??(x?B)
• ? ?x x?A?x?B

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Subset relations

• A?B
• ??(A?B?B?A)
• ??(A?B)??(B?A)
• ?(A B)? (B A)
• A?B
• ?A?B ?A?B

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Ex 3.5
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Theorems 3.1. and 3.2

• Theorem 3.1
• If A?B and B?C, then A?C,
• If A?B and B?C, then A?C,
• If A?B and B?C, then A?C,
• If A?B and B?C, then A?C,
• Theorem 3.2
• ?? A. If A is not empty, then ??A.

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Power set

• For any finite set A with ? A ?n, the total
  number of subsets of A is 2n.
• Definition 3.4. the power set of A, denoted as
  ?(A) is the collection of all subsets of A.
• What is the power set of 1, 2,3 4?

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Ex 3.10

• Count the number of paths in the xy-plane from
  (2,1) to (7,4)
• The number of paths sought here equals the number
  of subsets A of 1,2,,8, where ? A ?3.

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Ex 3.11

• Count the number of compositions of an integer,
  say 7
• 71111111, there are six plus signs.
• Subset 1,4,6 ? (11)1(11)(11)?2122
• Subset 1,2,5,6? (111)1(111)?313
• Subset 3,4,5,6? 11(11111)?115
• Consequently, there are 2m-1 compositions for the
  value m.

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An important identity

• C(n1, r) C(n, r)C(n, r-1)
• Pascala triangle in Ex 3.14

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3.2 Set operations and the laws of set theory

• Definition 3.5.
• A?Bx?x?A ? x?B
• A?Bx?x?A ? x?B
• A?Bx?x?A?B ? x?A?B
• Ex 3.15
• Definitions 3.6, 3.7, 3.8
• S, T are disjoint, written S?T?
• The complement of A, denoted as
• The relative complement of A in B, denoted B-A

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Ex 3.18
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Theorem 3.4

• The following statements are equivalent
• A?B
• A?BB
• A?BA

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A?B ? A?BB ?A?BA

• B?(A?B) for any sets
• x?(A?B) ? (x?A)?(x?B)
• since A?B, ? (x?B)
• this means (A?B)?B
• we conclude A?BB

• A?A?B for any sets
• y?A? y?A?B (1)
• Since A?BB, (1)?y?B?y?(A?B)
• This means A?A?B
• we conclude AA?B

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A?BA ? ? A?B
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A?(B?C)(A?B)?(A?C)
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The Duality

• Definition 3.9 Let s be a statement dealing with
  the equality of two set expressions. The dual of
  s, denoted sd, is obtained from s by replacing
  (1) each occurrence of ? and U by U and ?,
  respectively and (2) each occurrence of ? and ?
  by ? and ?, respectively.
• Theorem 3.5. s is a theorem if and only if sd is
  also a theorem.        

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Three approaches to proof

• The first approach to prove a theorem is by
  element argument.
• The second is by Venn diagram, and
• the third is by membership table.

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Venn diagram to show
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Venn diagram to show
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membership table
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Membership table for A?(B?C)(A?B)?(A?C)
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Ex 3.20
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Ex 3.22
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1
2
3
A1,2, B2,3, A?B1,3, A3,4,
B1,4 A?B2, 4 B?A (A?B)
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Generalized DeMorgans Law
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3.3. Counting and Venn diagrams

• Finite sets A and B are disjoint if and only if
  ? A ? B ? ? A ? ? B ? , Figures 3.9 and 3.10
• Ex 3.25, If A and B are finite sets, then ?A?B?
  ?A? ?B?-?A?B? , Figure 3.11
• When U is finite, we have

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Ex 3.26

• How many gates have at least one of the defects
  D1, D2, D3? How many are perfect?
• Figure 3.12 and Figure 3.13. If A, B and C are
  finite sets, then ?A?B?C? ?A?
  ?B??C?-?A?B?-?A?C?-?B?C? ?A?B?C?
• When U is finite, we have

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3.4. A first world on probability

• Let ? be the sample space for an experiment. Each
  subset A of ?, including the empty subset, is
  called an event. Each element of ? determines an
  outcome. If ???n, then Pr(a)1/n and Pr(A)? A
  ?/n
• Ex 3.29, Ex 3.30, Ex 3.31
• Definition 3.11. For sets A and B, the Cartesian
  product of A and B is denoted by A?B and equals
  (a, b)?a? A , b ? B. We call the elements of
  A?B ordered pairs.

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Ex 3.33

• Suppose we roll two fair dice.
• Consider the following event
• A rolls a 6
• B The sum of dice is at least 7
• C Rolls an even sum
• D The sum of the dice is 6 or less
• What are P(A), P(B), P(C), P(D), P(A?B), P(C?D)?

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Examples

• Ex 3.35. If we toss a fair coin four times, what
  is the prob that we get two heads and two tails?
• Ex 3.36. Among the letters WYSIWYG, what is the
  prob that the arrangement has both consecutive
  Ws and Ys? and the prob that the arrangement
  starts and ends with W?

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3.5. The axioms of probability

• Ex 3.39. The outcomes of a sample space may have
  different likelihoods
• A warehouse has 10 motors, three of which are
  defective. We select two motors.
• A exactly one is defective
• B at least one motor is defective
• C both motors are defective
• D Both motors are in good condition.

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The axioms of probability

• Let ? be the sample space for an experiment. If A
  and B are any events, then
• Pr(A)?0
• Pr(?)1
• If A and B are disjoint, Pr(A ? B )Pr(A) Pr(B)
  
• Theorem 3.7.

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Ex 3.40

• The letters PROBABILITY are arranged in a random
  manner. Determine the prob of the following
  event The first and last letters are different.
• Neither B nor I appears at the start or finish.
• (7)(9!/2!2!)(6)
• Only B appears at the start or finish.
• (2)(7)(9!/2!)
• One of B is used at the start and I as the other.
• (2)(9!)

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Ex 3.41

• The prob that our team can win any tournament is
  0.7. Suppose we need to play eight tournaments.
  Consider the following cases
• Win all eight games. (0.3)8
• Win exactly five of the eight. C(8,
  5)(0.7)5(0.3)3
• Win at least one. 1-(0.3)8
• If there are n trials and each trial has
  probability p of success and 1-p of failure, the
  probability that there are k successes among
  these n trials is

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Theorem 3.8

• Pr(A?B)
• Pr(A?Bc) Pr(B)
• Pr(A) Pr(B)- Pr(A?B )
• Ex 3.42
• What is the prob that the card drawn is a club
  and the value is between 3 and 7.
• Ex 3.43

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Theorem 3.9.

• Pr(A?B?C) Pr(A) Pr(B)Pr(C)-Pr(A?B)-Pr(A?C)-Pr(B
  ?C) Pr(A?B?C)