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Chapter 3 Set Theory

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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
- CD a subset, CD a proper subset
- C=D, two sets are equal
- Neither order nor repetition is relevant for a general set
- null set, {},

- AB
x [xAxB]

- A B
x [xAxB]

x [xAxB]

x [(xA)(xB)]

x [xA(xB)]

x [xAxB]

- AB
(ABBA)

(AB)(BA)

(A B) (B A)

- AB
AB AB

- Theorem 3.1
- If AB and BC, then AC,
- If AB and BC, then AC,
- If AB and BC, then AC,
- If AB and BC, then AC,

- 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.
- AB={xxA xB}
- AB={xxA xB}
- AB={xxAB xAB}
- Ex 3.15

- Definitions 3.6, 3.7, 3.8
- S, T are disjoint, written ST=
- The complement of A, denoted as
- The relative complement of A in B, denoted B-A

- The following statements are equivalent
- AB
- AB=B
- AB=A

B(AB) for any sets

x(AB) (xA)(xB)

since AB, (xB)

this means (AB)B

we conclude AB=B

AAB for any sets

yA yAB (1)

Since AB=B, (1)yBy(AB)

This means AAB

we conclude A=AB

- 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.

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)´

- 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 AB= A+ B-AB , 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 ABC= A+ B+C-AB-AC-BC + ABC
- 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 AB and equals {(a, b)a A , b B}. We call the elements of AB 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(AB), P(CD)?

- 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!)

- Neither B nor I appears at the start or finish.

- 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(AB)
=Pr(ABc) + Pr(B)

= Pr(A) + Pr(B)- Pr(AB )

- 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(ABC)= Pr(A)+ Pr(B)+Pr(C)-Pr(AB)-Pr(AC)-Pr(BC) + Pr(ABC)