Title: Chapter 3 Set Theory
1Chapter 3Set Theory
- Yen-Liang Chen
- Dept of Information Management
- National Central University
23.1 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, , ?
3Subset 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
4Subset relations
- A?B
- ??(A?B?B?A)
- ??(A?B)??(B?A)
- ?(A B)? (B A)
- A?B
- ?A?B ?A?B
5Ex 3.5
6Theorems 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.
7Power 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?
8 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.
9Ex 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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11An important identity
- C(n1, r) C(n, r)C(n, r-1)
- Pascala triangle in Ex 3.14
123.2 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
13Ex 3.18
14Theorem 3.4
- The following statements are equivalent
- A?B
- A?BB
- A?BA
-
15A?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
16A?BA ? ? A?B
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19A?(B?C)(A?B)?(A?C)
20The 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.
21Three 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.
22Venn diagram to show
23Venn diagram to show
24 membership table
25Membership table for A?(B?C)(A?B)?(A?C)
26Ex 3.20
27Ex 3.22
284
1
2
3
A1,2, B2,3, A?B1,3, A3,4,
B1,4 A?B2, 4 B?A (A?B)
29Generalized DeMorgans Law
303.3. 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
31Ex 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
323.4. 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.
33Ex 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)?
34Examples
- 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?
353.5. 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.
36The 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.
37Ex 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!)
38Ex 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
39Theorem 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
40Theorem 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)