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# Chapter 3 Set Theory - PowerPoint PPT Presentation

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 3Set Theory

Yen-Liang Chen

Dept of Information Management

National Central University

• 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, {}, 

• 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]

• AB

(ABBA)

(AB)(BA)

(A B) (B A)

• AB

AB AB

• 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.

• 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}?

• 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.

• 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.

• C(n+1, r)= C(n, r)+C(n, r-1)

• Pascal’a triangle in Ex 3.14

• 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

• The following statements are equivalent

• 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C)=(AB)(AC)

• 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.

• The first approach to prove a theorem is by element argument.

• The second is by Venn diagram, and

• the third is by membership table.

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

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)´

• 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

• 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

• 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.

• 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)?

• 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?

• 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.

• 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.

• 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!)

• 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

• 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

• Pr(ABC)= Pr(A)+ Pr(B)+Pr(C)-Pr(AB)-Pr(AC)-Pr(BC) + Pr(ABC)