# 1 Chapter 3 Set Theory Yen-Liang Chen Dept of Information Management National Central University.

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1 Chapter 3 Set Theory Yen-Liang Chen Dept of Information Management National Central University

2 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, {}, 

3 Subset relations  AB  x [xAx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)]  x [xAxB]

4 Subset relations  AB (ABBA) (ABBA) (AB)(BA) (A B) (B A)  AB AB AB AB AB

5 Ex 3.5

6 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.

7 Power set  For any finite set A with  A =n, the total number of subsets of A is 2 n.  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 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.

9 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 2 m-1 compositions for the value m.

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11 An important identity  C(n+1, r)= C(n, r)+C(n, r-1)  Pascal ’ a triangle in Ex 3.14

12 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

13 Ex 3.18

14 Theorem 3.4  The following statements are equivalent –AB –AB=B –AB=A –

15 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

16 A  B=A   A  B

17

18

19 A  (B  C)=(A  B)  (A  C)

20 The Duality  Definition 3.9 Let s be a statement dealing with the equality of two set expressions. The dual of s, denoted s d, 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 s d is also a theorem.  Theorem 3.5. s is a theorem if and only if s d is also a theorem.

21 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.

22 Venn diagram to show Venn diagram to show

23 Venn diagram to show

24 membership table membership table

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

26 Ex 3.20

27 Ex 3.22

28 1 2 3 4 A={1,2}, B={2,3}, A∆B={1,3}, A´={3,4}, B´={1,4} A´∆B={2, 4}= B´∆A = (A∆B)´

29 Generalized DeMorgan ’ s Law

30 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

31 Ex 3.26  How many gates have at least one of the defects D 1, D 2, D 3 ? 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

32 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.

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)?

34 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?

35 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.

36 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.

37 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!)

38 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

39 Theorem 3.8  Pr(A  B) =Pr(A  B c ) + Pr(B) =Pr(A  B c ) + Pr(B) = Pr(A) + Pr(B)- Pr(A  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

40 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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