1. Chapter Zero A Review of Set Notation
STAT 111
Chapter Zero
A Review of Set Notation

2. A Review of Set Notation
Definition:
A set is a well-defined collection (possible empty) of objects, such as the set of letters in the alphabet, or the set of students in a classroom. We denote a set by a capital letter such as A,B,C and an element by a lower case letter such a,b,c.
Notes:
If an element a belongs to a set C we write a ϵ C.
If an element a does not belong to C we write a ϵ C.

3. Presentation
Sets may be described by listing the elements or by describing some property held by all members. For example the set P of positive integers might be described by:
P={1,2,3,…}
P={x : x is a positive integer}

4. Example
Let A be the set of all real numbers whose squares are equal to 25. Show how to describe A by
Listing its elements,
A={-5,+5} since x2= x=-5 or x=+5
By describing some property held by all members.
A={X : X2=25}

5. Example
Determine which of the following statement are true and which are false.
{2}=2
False (since 2 is areal number while {2} is a set which consists of the real number 2)
True (since every elements is assumed to be equal to itself)
True (since no number satisfy which is a subset of every set

6. Notations and Venn Diagram
The universal set S is the set of all elements under consideration. Using Venn diagram it consists of a rectangle.
The null or empty set, denoted by Φ, is the set consisting of no elements. Thus Φ is a subset of every set. A B
. For any two sets A and B we will say that A is
a subset of B, or A is contained in B
(denoted A B), if every element in A is also in B.
Remarks:
Every set is a subset of itself
If AB and BA then A=B

7. Notations and Venn DiagramB S A B
A U B
A ∩ B
The union of A and B, denoted by A U B, is the set of all elements which belong to either A or B or both. That is, the union of A and B contain all elements that are in at least one of the sets.
The key word for expressing the union of two
sets is or (meaning A or B or both).
The intersection of A and B, denoted by A ∩ B,
is the set of all elements which belong to both
A and B. The key word for expressing the intersections is and (meaning A and B simultaneously).

8. Notations and Venn Diagram
If A is a subset of S , then the complement of A, denoted by Ac (or A ), is the set of elements that are in S but not in A. the shaded area in S but not in A is Ac. Note that A U Ac =S. Note also, (Ac)c=A S A Ac

9. Notations and Venn DiagramB S A B
Two sets, A and B, are said to be
disjoint or mutually exclusive
if A ∩ B=Ф. That is, mutually exclusive
sets have no elements in common.
Note that, it is easy to see that A and Ac
are mutually exclusive sets for any set A.
The difference of A and B, denoted by A-B,
is the set of all elements of A which do
not belong to B. Note that
A-B=A ∩ Bc
Note also A=(A∩B)U(A∩Bc),
and, B=(B∩A)U(B∩Ac)

10. Example
If S= {1/2,0,л,5, - √ 2 , - 4 } and subsets of S are given by
A={- √2, л,0}, B={ 5, ½,-√ 2,- 4} and C={ 1/2,-4 } find
A ∩ B, A U B,(A U B ) ∩ C, A- B, ( B ∩ C )c, Bc U Cc and (A ∩ C ) U ( B ∩ C )
Solution:
A ∩ B={- √ 2 }
A U B= {1/2,0,л ,5, - √ 2 , - 4 } (A U B ) ∩ C ={ 1/2,-4 }
A-B={л,0}
Bc={л,0} Cc={0,л ,5, - √ 2} Bc U Cc ={0,л,5, - √ 2} = ( B ∩ C )c
(A ∩ C ) U ( B ∩ C )=Φ U { 1/2,-4 } = { 1/2,-4 }

11. Example n (LU U)=n(L)+n(U)-n(L∩ U)=9+36-3=42
From a survey of sixty students attending a university, it was found that nine were living off campus, thirty-six were undergraduates and three were undergraduates and living off campus.
Find the number of these students who were undergraduates, or living off campus, or both.
n (LU U)=n(L)+n(U)-n(L∩ U)=9+36-3=42

12. S c) Find the number of these students who
Find the number of these students who were undergraduates and living on campus.
c) Find the number of these students who
were graduate students living on campus.
n(Uc ∩ Lc)=18 L Lc
Sum U 3 36 Uc 9 60 33 6 18 24 51 S L U 6 33 18 3
n(U ∩ Lc)=33

13. Example
يوجد في معهد الامل للصم 300طالب يدرسون لغة الاشارة بالانجليزية أو بالعربية, 150يدرسون لغة الاشارة بالعربية فقط, 80يدرسون لغة الاشارة بالانجليزية فقط.
ما هو عدد الذين يدرسون لغة الاشارة بالانجليزية والعربية؟
ما هو عدد الذين يدرسون لغة الاشارة بالعربية؟
ما هو عدد الذين يدرسون لغة الاشارة بالانجليزية؟
ما هو عدد الذين لا يدرسون لغة الاشارة بالانجليزية ولا بالعربية؟ S E A 80
150 70

14. Some Theorems Involving Sets
1)A U B = B U A (commutative law for unions)
2) A ∩ B = B ∩ A (commutative law for inter sections)
3) A U ( B U C ) = (A U B ) U C = (associative law for unions)
4)A ∩ (B ∩ C)= (A∩B)∩C = A ∩ B ∩ C (associative law for inter section
5) A∩( B U C) = (A ∩ B)U (A∩C) (first distributive law)
6) A U (B ∩ C) = (A U B)∩ (A U C) (second distibutive law)
7) A -B = A ∩ Bc
8) if A  B then Bc  Ac
9) A U Ø = A , A ∩ Ø = Ø
10 ) A U S = S , A ∩ S = A
11) De Morgran's law
a) ( A U B )c = A c ∩ B c
b) (A ∩ B ) c = A c U B c (De Morgan's laws can be generalized to any relation between
For Example A ∩ B c = ( A c U B ) c , and (A c U B) c = A ∩ B c
12 ) A = ( A ∩ B ) U ( A ∩ B c)

15. Generalizations of Definitions
The union of the sets A1,A2,A3,…, denoted
A1 U A2 U A3 ,… is the set of all elements which are in at least one member of the collection.
The intersection of the sets A1,A2,A3,…, denoted,
A1∩A2 ∩A3,.. is the set of all elements which are in every member of the collection. \
The collection A1,A2,A3,…, is a pairwise disjoint if
A i ∩ A j = Ф whenever i ≠ j

16. Generalizations of Some Theory Involving Sets
1-B ∩ (A1 U A2 U A3 …) = (B ∩ A1 ) U ( B ∩ A2) U (B ∩ A3 ) …
2-B U (A1∩A2 ∩A3 …)= ( B U A1) ∩ ( B U A2) ∩ (B U A3) ∩ …
3-De Morgan's laws

17. Cartesian Products
The Cartesian product of the sets A1, A2,… ,An, denoted by A1 A2  …  An, is the set of all n-tuples with the ith coordinate drawn from the ith set, i=1,2,…,n, i.e.,
A1 A2  …  An={(a1, a2,… ,an): ai ϵ Ai, i=1,2,…,n}
Example
let A1 = {1,2,3}, , A2 = {a , b }, then
A1 × A2 = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}
وعدد العناصر لحاصل الضرب هو
n (A1 × A2 ) = n (A1) × n(A2 )
The Cartesian product may be plotted in 2-dimensional space using points on the horizontal axis for the elements of A1, and points of the vertical axis for the elements of A2, the resulting ordered pairs represent

18. Power Sets
The power set of A is the set of all subsets which may be formed from the elements of A .
A جميع المجموعات الجزئية المكونة من
Example
Let A= { a , b , c} . Find the power set of A
Power set of A= { Ø , A , {a } , { b } , { c } , {a,b} ,
{a ,c } , { b ,c}}

19. Chapter One Combinatorics
STAT 111
Chapter One
Combinatorics

20. Combinatorics
In many experiment the number of outcomes in S is so large that a complete listing of these outcomes is too difficult. In such experiment, it is convenient to have a mathematical method of determining the total number of outcomes in the space S and in various events in S without compiling a list of all these outcomes. In the following, some of these methods will be presented.
طرق العد هي مجموعة من الطرق الرياضية التي تساعد علي معرفة عدد مرات حدوث تجربة ما دون الحاجة لكتابة عناصر هذه المجموعة

21. The Basic Principle of Couting (Multiplication Rule )
If an experiments consists of r steps , of which the first can be made in n1 ways , for each of these the second step can be made in n2 ways , and so forth , then the whole experiment can be made in
n1 × n2 × …× nr

22. The Basic Principle of Couting (Multiplication Rule )
Example 1:
A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. In how many different ways can a doctor prescribe the drug for a patient suffering from asthma
5x3x2 = 30
Example 2:
In how many ways can 8 teaching assistants be assigned to 6 sections of a statistic course if no teacher is assigned to more than one section? x7x6x5x4x3 = 20160

23. The Basic Principle of Couting (Multiplication Rule )
Example 3:
How many three-digit numbers can be formed from the digits 0,1, 2,3,4,5, and 6 if each digit can be used only once (التكرار غير مسموح)
6x 6 x 5 = 180
b) How many of these are odd numbers?
5x5x3 = 75
How many are greater than 330?
1x3x5 + 3x6x5 = = 105
لايتم اختيار الصفر في هذه الخانة
الأعداد الأكبر من 3 وهي 6 ، 5 ، 4

24. The Basic Principle of Couting (Multiplication Rule )
Example 4: If a multiple-choice test consists of 5 questions each with 4 possible answers of which only one is correct,
How many different ways can a student check off one answer to each question ?
4 × 4 × 4 × 4 × 4 = ( 4 )5 = 1024
How many ways can a student check off one answer to each question and get all the question wrong ?
3× 3 × 3 × 3 × 3 = ( 3 ) 5 = 243

25. Permutations
Frequently, we are interested in situations where the outcomes are the different orders or arrangements that are possible for a group of objects. Different arrangements like these are called permutations.
Note:
n!=n(n-1)(n-2)…3x2x1
0!=1
Definition
A permutation is an ordered arrangement of a set of distinct objects. The number of permutations of n distinct objects taken r at a time is denoted by where
=n! / (n-r)!= n(n-1) (n-2)….(n-r+1)
التباديل هي عدد طرق الاختيار عند توفر شرطين
الترتيب مهم ( جوائز مختلفة مراكز مختلفة )
التكرار غير مسموح

26. Examples
Example
A president, treasurer (أمين صندوق), and a secretary, all different, are to be chosen from a club consisting of 10 people. How many different s of officers are possible?
Example
Five separate awards are to be presented to selected Students from a class of 30 . How many different outcomes are possible if A student can receive any number of awards ;
Each student can receive at most 1 award?

27. notes
In if r= n then
which represents the different ways to arrange n objects in a line.
The number of different ways to arrange n objects in a circle is
( n – 1 ) !

28. Examples
In how many different ways can the five letters a, b, c, d and e be arranged?
5! = 120
In how many ways can 3 Arabic books, 2 Mathematics books, and 1 Chemistry book be arranged on a bookshelf if;
The books can be arranged in any order;
6! = 720
the Arabic books must be together but the other books can be arranged in any order ;
3! 4! = 144
the Mathematics books must be together and the Arabic books must be together ?
3 ! 3 ! 2 ! = 72

29. Example 3. a) In how many ways can 5 persons line up to get on a bus.
5! = 120
b) In how many ways can they line up, if 3 persons insist on following each other?
3! 3! =36
c) In how many ways can they line up, if 3 persons refuse to follow each other
n(A) = n(S)-n(Ac) ! 3! = 84
4. a) In how many ways may four women and four children be seated in a row of chairs if the women and children to occupy alternate seats'?
2 (4 x 4 x 3 x 3 x 2 x 2 x l x l) = 2 (4! 4!)
b) Repeat a) when they are seated at a round table.
(4-1)! 4!
5. In how many ways can 7 people be seated at a round table if
a) They can sit anywhere;
(7-1)! = 6! = 720
b)Two particular people must not sit next to each other?
6! - 2! 5! = 480

30. Permutations When Some Elements Are Alike
Given n objects of which n1 are alike, n2 are alike,..., nr are alike, then the total number of permutations is
Example 1
How many different ways can we arrange the letters in the word
Tool
4!/2!
Statistics
10!/(3!3!2!)
Example 2
How many different arrangement are there of the following set {a , α , α , β , β , γ } .
6!/(3!2!)

31. Combinations
In many problems we are interested in the number of selecting r objects from
n without regard to order (the order is unimportant), these selections are called combinations.
التوافيق هي عدد طرق الاختيار عندما يكون
الترتيب غير مهم
التكرار غير مسموح
Definition
The number of combinations of r objects is
The number is also denoted by the symbol Note that represents the number of ways we can draw a sample of size r from a set of n distinct elements without replacement and without regard to the order in which these r elements were chosen. It is defined that = 0 whenever n is a positive integer and r is a positive integer greater than n (clearly, there is no way in which we can select a subset which contains more elements than the set itself if r < 0 or r > n ) الفرق بين التباديل والتوافيق هو الترتيب
( إذا تم اختيار شخصين من A, B, C
الترتيب مهم
AB , BA , AC , CA , BC , CB
AB , AC , BC

32. Examples
Example 1
In how many different ways can a committee of 4 be selected from among the 64 staff members of a hospital?
Example 2
A committee of 7, consisting of 2 doctors, 2 teachers, and 3 engineers, is to be chosen from a group of 5 doctors, 6 teachers, and 4 engineers. How many committees are possible?

33. Examples
Example 3
From a group of 4 teachers and 5 doctors, how many committees of size 3 are possible;
with no restrictions;
with 1 teacher and 2 doctors;
With 2 teachers and 1 doctor if a certain teacher must be on the committees;
with at least 1 teacher:
with at most 2 doctors?

34. Examples
From a group of 8 statisticians and 6 economists a committee consisting of 3 statisticians and 3 economists is to be formed. How many different committees are possible if
1- Two of the statisticians refuse to work together.
2- Two of the economists will only work together.
3- One statistician and one economist refuse to work together.

35. Theorem
for any positive integers n and r = 0, 1... n.
Result:

36. The Multinomial Theorem
The number of possible divisions of n distinct objects into r distinct groups elements in the first group, n2 elements in the second, and so forth, is
جميع الطرق الممكنة لتقسيمn من الأشياء أو الأشخاص إلي
rمن المجموعات بحيث تحتوي علي
من العناصر n 1 , n2 , … ,nr

37. Multinomial
Example 1:
If 12 People are to be divided into 3 committees of respective sizes 3,4 and , 5 how many divisions are possible ?
In how many ways can a man divide 7 gifts among his 3 children if the eldest to receive 3 gifts and the other 2 each ?

38. The Binomial Theorem
The number is often referred to as the binomial coefficient because it arises in the binomial theorem which may be started as follow : For any number x and y and any positive integer n ,

39. Examples
Example 1 Expand
Example 2 Expand
Example 3 Find the constant term in the expansion of
Since constant term correspond to the one with 3 k – 12 =0 , therefore , K = 4 and the constant term

40. Example 4 Find the constant term in the expansion of
Since constant term correspond to the one with 3 k – 12 =0 , therefore , K = 4 and the constant term