Chapter 3 set theory
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Chapter 3 Set Theory. Yen-Liang Chen Dept of Information Management National Central University. 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 C = D , two sets are equal

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Chapter 3 Set Theory

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Chapter 3 set theory

Chapter 3Set Theory

Yen-Liang Chen

Dept of Information Management

National Central University


3 1 sets and subsets

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

    • C=D, two sets are equal

    • Neither order nor repetition is relevant for a general set

    • null set, {}, 


Subset relations

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]


Subset relations1

Subset relations

  • AB

    (ABBA)

    (AB)(BA)

    (A B) (B A)

  • AB

    AB AB


Ex 3 5

Ex 3.5


Theorems 3 1 and 3 2

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.


Power set

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}?


Ex 3 10

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.


Ex 3 11

Ex 3.11

  • Count the number of compositions of an integer, say 7

  • 7=1+1+1+1+1+1+1, there are six plus signs.

    • Subset {1,4,6}  (1+1)+1+(1+1)+(1+1)2+1+2+2

    • Subset {1,2,5,6} (1+1+1)+1+(1+1+1)3+1+3

    • Subset {3,4,5,6} 1+1+(1+1+1+1+1)1+1+5

  • Consequently, there are 2m-1 compositions for the value m.


An important identity

An important identity

  • C(n+1, r)= C(n, r)+C(n, r-1)

  • Pascal’a triangle in Ex 3.14


3 2 set operations and the laws of set theory

3.2 Set operations and the laws of set theory

  • Definition 3.5.

    • AB={xxA  xB}

    • AB={xxA  xB}

    • AB={x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


Ex 3 18

Ex 3.18


Theorem 3 4

Theorem 3.4

  • The following statements are equivalent

    • AB

    • AB=B

    • AB=A


A b a b b a b a

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B=B

AAB for any sets

yA yAB (1)

Since AB=B, (1)yBy(AB)

This means AAB

we conclude A=AB

AB AB=B AB=A


A b a a b

AB=A   AB


A b c a b a c

A(BC)=(AB)(AC)


The duality

The Duality

  • Definition 3.9Let 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.


Three approaches to proof

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.


Venn diagram to show

Venn diagram to show


Venn diagram to show1

Venn diagram to show


Membership table

membership table


Membership table for a b c a b a c

Membership table for A(BC)=(AB)(AC)


Ex 3 20

Ex 3.20


Ex 3 22

Ex 3.22


Chapter 3 set theory

4

1

2

3

A={1,2}, B={2,3}, A∆B={1,3}, A´={3,4}, B´={1,4}

A´∆B={2, 4}= B´∆A = (A∆B)´


Generalized demorgan s law

Generalized DeMorgan’s Law


3 3 counting and venn diagrams

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


Ex 3 26

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


3 4 a first world on probability

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.


Ex 3 33

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)?


Examples

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 W’s and Y’s? and the prob that the arrangement starts and ends with W?


3 5 the axioms of probability

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.


The axioms of probability

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.


Ex 3 40

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!)


Ex 3 41

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


Theorem 3 8

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


Theorem 3 9

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)


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