BLACKBOXTAF 3023:

MATHEMATICS DISCRET

TASK 2:CONCEPT OF SET

DEFINITIONS OF SETS

• A set is any well-defined collection of objects• The elements or members of a set are the objects contained in the set• Well-defined means that it is possible to decide if a given object belongs to the collection or not.

1)WAY OF LISTING THE ELEMENTS OF SETS

A. Roster Method -listing the elements in any order and

enclosing them with braces.Example: A= {January, February, March…

December} B={1,3,5…}

B. Set builder notation -giving a descriptive phrase that will clearly

identify the elements of the set. Example: { x Є A| P(x) }

2)BASIC PROPERTIES OF SETS sets are inherently unordered

-no matter what a,b, and c denoted {a,b,c}={a,c,b}={b,a,c}= {b,c,a}={c,a,b}={c,b,a} All element are unequal -if a=b then {a,b,c}={a,c}= {b,c}={a,a,b,a,b,c,c,c,c} -this sets contains at most 2 elements -number of occurence are not important)

3)SET MEMBERSHIP• ϵ-- “is an element of”(note that it is a

shape like “E” as in an element).• ∉-- “is not an element of”.• Example:-If A={4,7,5,8}, then 4 ϵ A, but 2

∉ A.

4)EMPTY SET• ᴓ is the label for empty set.• Example:- ᴓ ={ }

5)SET OF NUMBER• In term of capital alphabet to show the

specific set of number.

• Example: 1) natural number, N= { x:x = 1,2,3….}2) Integer, Z = {x:x = …-2,-1,0,1,2}3) Rational number, Q { x:x = is a rational

number}4)Irrational number, R { x:x = irrational

number}

• In term of capital alphabet to show the specific set of number.

• Example: 1) natural number, N= { x:x = 1,2,3….}2) Integer, Z = {x:x = …-2,-1,0,1,2}3) Rational number, Q { x:x = is a rational

number}4)Irrational number, R { x:x = irrational

number}

6)SET OF EQUALITY• Set which has same or equal element in term on it’s

declaration in the content.

• Note that set have a same element

• In some of case, it does not matter if the arrangement is correct order or not

How to understand ( if the set is equal)

• Let’s assume that set P is defined same set of Q.

• So, if the set have same element and although not in correct order it can be considered the both set is equal.

How to understandHow to understand(if it is not equal)(if it is not equal)

• If the set P does not have same element with set Q , so it is not equal.

• Eventhough the least one element either set P or set Q not in both set, it consider as not equal laaa…

• Example :P = {a, b, c, d} Q = {d, c, b, a}  P = Q each elements of set P that is a, b, c, d is same

with  element of set Q that is 8, 4, 2, 6. rearrange the elements of the set it still be same.

7)VENN DIAGRAMalsoalso knownknown asas setset diagramdiagram. . whichwhich knownknown asas aa diagramdiagram indicatesindicates allall ofof

thethe possiblepossible logicallogical relationsrelations betweenbetween setssets ofof finitefinite collectioncollection ((aggregationaggregation).).

VennVenn diagramsdiagrams areare alternativesalternatives toto solvesolve problemsproblems inin marketmarket research,research, inin science,science, inin socialsocial science,science, etcetc. .

wherewhere oftenoften overlappingoverlapping informationinformation isis obtainedobtained andand needsneeds toto bebe organizedorganized..

historyhistory factsfacts onon VennVenn DiagramDiagram::

thethe VennVenn DiagramDiagram werewere formulatedformulated byby JohnJohn VennVenn onon 18801880. . Nowadays,Nowadays, TheThe VennVenn DiagramDiagram cancan bebe useuse toto illustrateillustrate simplesimple setset relationshiprelationship inin

probability,probability, logic,logic, statistics,statistics, linguistics,linguistics, andand computercomputer sciencescience..

examplesexamples

1. From a survey of 100 workers, a marketing

research company found that 75 workers owned

stereos, 45 owned cars, and 35 owned cars and

stereos.

a) How many students owned either a car or a stereo?

b) How many students did not own either a car or a stereo?

1. From a survey of 100 workers, a marketing

research company found that 75 workers owned

stereos, 45 owned cars, and 35 owned cars and

stereos.

a) How many students owned either a car or a stereo?

b) How many students did not own either a car or a stereo?

METHOD?METHOD?

a) Start with a Venn Diagram and label the different categories:

b) Fills the number of workers who own both cars and stereos, resulting in the intersection of the two sets:

c)Fills the remaining numbers for the both 2 sets. So, the total of 45 workers own cars, and 35 have already been listed, then 45 - 35 = 10 workers own cars only.Same goes to the 75 workers own stereos and 35 point out.Thus, 75 - 35 = 40 workers who own stereos only:

a) Start with a Venn Diagram and label the different categories:

b) Fills the number of workers who own both cars and stereos, resulting in the intersection of the two sets:

c)Fills the remaining numbers for the both 2 sets. So, the total of 45 workers own cars, and 35 have already been listed, then 45 - 35 = 10 workers own cars only.Same goes to the 75 workers own stereos and 35 point out.Thus, 75 - 35 = 40 workers who own stereos only:

answers!answers!therefore, after some footages on the diagrams drawn. We are able to solve the questions!

a) How many workers have either a car or a stereo?The question asks either or which is union of the sets.From what we able to gain from the diagram, the number of elements in A = 10 + 35and the number of elements in B which are NOT in A (different area) are 40.So the union would be 10 + 35 + 40 = 85

b) How many workers did not have either a car or a stereo?

The question asks for the number not in either A nor B(namely, the complement of A B or (A B)' ).Since there are 100 workers assumed, then the total is obtainedby subtracting those who own either a car or stereo from the total number of workers surveyedin such expression:

100 - 85 = 15.

therefore, after some footages on the diagrams drawn. We are able to solve the questions!

a) How many workers have either a car or a stereo?The question asks either or which is union of the sets.From what we able to gain from the diagram, the number of elements in A = 10 + 35and the number of elements in B which are NOT in A (different area) are 40.So the union would be 10 + 35 + 40 = 85

b) How many workers did not have either a car or a stereo?

The question asks for the number not in either A nor B(namely, the complement of A B or (A B)' ).Since there are 100 workers assumed, then the total is obtainedby subtracting those who own either a car or stereo from the total number of workers surveyedin such expression:

100 - 85 = 15.

8)SUBSET

• Set A is a subset of set B if and only if every element in A is also in B.

• Example : We have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set

• NOTATION : When we say that A is a subset of B, we write A B. Or we can say that A is not a subset of B by A B ("A is not a subset of B")

Example: Let A be multiples of 4 and B be multiples of 2. Is A a subset of B? And is B a subset of A?By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:

•So, A is a subset of B, but B is not a subset of A.

Proper Subsets•A is a proper subset of B if and only if every element in A is also in B, and there exists at least one element in B that is not in A.•Example: {1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}. •Example: {1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set. •You should notice that if A is a proper subset of B, then it is also a subset of B.

9)POWER SET

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•  

10)SET OPERATIONA)UNIONunion of the sets A and B, denoted by A ∪ Belements that are either in A or in B, or in both.A ∪ B = {x | x ∈ A ∨ x ∈ B}.

B)INTERSECTION• intersection of the sets A and B, denoted by A ∩

B• set• containing those elements in both A and BA ∩ B = {x | x ∈ A ∧ x ∈ B}.

C)DISJOINT SET• Two sets are called disjoint if their intersection

is the empty set.• Meaning that two sets has nothing in common

D)SET DIFFERENCE• Definition• The (set) difference between two se

ts S and T is written S∖T, and means the set that consists of the elements of S which are not elements of T:

• x∈S∖T⟺x∈S∧x∉TIt can also be defined as:

• S∖T={x∈S:x∉T}• S∖T={x:x∈S∧x∉T}

• Illustration by Venn Diagram

The red area in the following Venn diagram illustrates S∖T:

• Example

For example, if S={1,2,3} and T={2,3,4}, then S∖T={1}, while T∖S={4}.

It can immediately be seen that S∖T is not commutative, in general (and in fact, that it is anticommuntative).

E)SET COMPLEMENTARYSo far, we have considered operations in which two sets combine to form a third: binary operations. Now we look at a unary operation - one that involves just one set.The set of elements that are not in a set A is called the complement of A. It is written A′ (or sometimes AC, or ).Clearly, this is the set of elements that answer 'No' to the

question Are you in A?.For example, if U = N and A = {odd numbers}, then A′ = {even numbers}.

F)CHARACTERISTIC OF SETS

11)GENERALISED UNION AND INTERSECTION

• Union• A ∪ B ∪ C contains those elements that are in at

least one of the sets A, B, and C• parentheses do not use to indicate which

operation comes first• A ∪ (B ∪ C) = (A ∪ B) ∪ C

• Intersection• A ∩ B ∩ C contains those elements that are

in all of A, B, and C• parentheses also do not use to indicate

which operation comes first• A ∩ (B ∩ C) = (A ∩ B) ∩ C

• Example:

• Let A = {0, 2, 4, 6, 8}, B = {0, 1, 3, 5, 7} and C = {0, 1, 2, 3}. What are A ∪ B ∪ C and A ∩ B ∩ C ?

• Solution:

The set A ∪ B ∪ C contains those elements in at least one of A, B, and C.

• A ∪ B ∪ C = {0, 1, 2, 3, 4, 5, 6, 7}

The set A ∩ B ∩ C contains those elements in all three of A, B, and C.

• A ∩ B ∩ C = {0}

• Generalized Union• The union of a collection of sets is the set that contains those

elements that are members of at least one set in the collection.

• Using notation:

to donate the union of the sets A₁, A₂, . . . , An.• Generalized Intersection• The intersection of a collection of sets is the set that contains

those elements that are members of all the sets in the collection.

• Using notation:

• to denote the intersection of the sets A₁, A₂, . . . , An.

12)CARTESIAN PRODUCT