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Chapter 3 Set Theory

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3.1 Sets and subsets

- 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, {},

Subset relations

- AB
x [xAxB]

- A B
x [xAxB]

x [xAxB]

x [(xA)(xB)]

x [xA(xB)]

x [xAxB]

Subset relations

- AB
(ABBA)

(AB)(BA)

(A B) (B A)

- AB
AB AB

Theorems 3.1. and 3.2

- 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.

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

- 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

- 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

- 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

- 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

Theorem 3.4

- 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

AB AB=B AB=AAB=A AB

A(BC)=(AB)(AC)

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

- 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(BC)=(AB)(AC)

Generalized DeMorgan’s Law

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 AB= A+ B-AB , Figure 3.11
- When U is finite, we have

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 ABC= A+ B+C-AB-AC-BC + ABC
- When U is finite, we have

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 AB and equals {(a, b)a A , b B}. We call the elements of AB ordered pairs.

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(AB), P(CD)?

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

- 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

- 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

- 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.

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

- 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

Theorem 3.9.

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