chapter 2:concept of set
TRANSCRIPT
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DISCRETE MATHEMATICS
FORESPEC MEMBER:
Nor Farahana Zainul Hisham (032291)
Nurul Syazwani Kamarulzaman(032691)Nor Shila Latif (032840)
Nor Farahin Rosli (032963)
Nor Irma Fariza Mohd Zamri (033053)
Sam Tau Siong (032261)
Muhammad Husni Ideris (032500)
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CHAPTER 2 :CONCEPT OF SET
Way of listing the elements of Sets
Specifying properties of sets
Set membership
Empty set
Set of numbers (Z,N,etc....)
Set Equality Venn Diagram
Subset
Power Set
Set Operation
Generalised Union and Intersection
Cartesian Product
Second Task: TAF3023
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Way of listing the elements of Sets
1. A written description (Extension)
Example:
A is the set of calendar months beginning with the letter
2. List or Roster Method (Intentional)
Example:
A= { January, June, July}
3. Set builder notation
Example:
A= { x x is a calendar month beginning with the letter J}
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Example:
C is the set of primary colour
C = { Blue, Red, Yellow}
C = {xx is a primary colour}
Blue
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Properties of sets
Sets are inherently unordered: No matter what objects a, b, and c denote,
{a,b,c} ={a,c,b} ={b,a,c} = {b,c,a} = {c,a,b} = {c,b,a}.
All elements are distinct (unequal), multiple listings make nodifference
{a,a,b} = {a,b,b} = {a,b} = {a,a,a,a,b,b,b}.
This set contains at most 2 elements
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Set Membership
Set is an unordered collection of object, the objectinside the set, called elements or members.
We use symbol and to denote the element of
a set.
For example, aA mean a is the element of the set
A. While bA mean b is not the element of set
A.
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Empty Set
There is a special set, that are no element inside it,
which called empty set or null set.
There are 2 symbols to denote an empty set, and{}.
It's quite interested when these 2 symbols combine
together to become {} .It's not a empty set ,it'scalled singleton set. Which represent an empty set in
a set itself.
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Set of numbers
Natural numbers N ={0, 1, 2, 3,...},
Integers Z ={...,2,1, 0, 1, 2,...},
Positive integers Z+ ={1, 2, 3,...},Rational numbers Q ={p/q | p Z,q Z, and q = 0},
real numbers R
Positive real numbers R+,
complex numbers C
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Set Equality
Definition: Set Equality Two sets are equal if they are bothsubsets of one another. The standard notation for equality, =, is
used.
Example:
{1, 2, 3} = {3, 2, 1} = {1, 1, 2, 3, 2, 2}
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Venn Diagram
These are a pictorial representation of sets and a good way to getintuition about (possible)set equalities.
Universal set U= contains all the objects under consideration, is
represented by a rectangle.
Inside this rectangle, circles or other geometrical figures are usedto represent sets.
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Example Venn Diagram
You can also use Venn Diagrams for 3 sets.Let us say the third set is "Volleyball", which drew, glen and jade play:
Volleyball = {drew, glen, jade}
But let's be more "mathematical" and use a Capital Letter for each set:
S means the set of Soccer playersT means the set of Tennis players
V means the set of Volleyball players
The Venn Diagram is now like this:
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Example Venn Diagram
In a school of 320 students, 85 students are in the
band, 200 students are on sports teams, and 60students participate in both activities. How many
students are involved in either band or sports?
25 + 60 + 140 = 225
There are 225 students involved in either band or sports.
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Subset
Encounter situations where the elements of one setare also the elements of a second set.
If all the elements of a set A are also elements of a
set B, then we say that A is a subsetofB, and we
write:
AB
For example:
IfA = {2, 4, 6, 8, 10} and B = {even integers},
thenAB
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Power Set
If we have a set {a,b,c}:
Then a subset of it could be {a} or {b}, or {a,c}, and so
on, and {a,b,c} is also a subset of {a,b,c} .
And the empty set {} is also a subset of {a,b,c}
In fact, if you list all the subsets of S={a,b,c} you willhave the Power Set of {a,b,c}:
P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
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Set OperationUnion
The union of a collection of sets is the set that
contains those elements that are members at least
one set in the collection.
The union of two sets is a new set which combines all
of the members of both sets (and discards duplicates).
Eg: IfA = {1, 2, 3} and B = {4, 5} ,
thenA B = {1, 2, 3, 4, 5} .
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IfA = {1, 2, 3} and B = {4, 5} ,
thenA B = {1, 2, 3, 4, 5} .
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2
3
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5
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Set Operation
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.
The intersection of two sets is a new set which
only includes those members present in both
sets.
Eg: IfA = {1, 2, 3} and B = {1, 2, 4, 5} ,
thenAB = {1, 2} .
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IfA = {1, 2, 3} and B = {1, 2, 4, 5} ,
thenAB = {1, 2} .
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23
4
5
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Set Operation
Disjoint Set
Intersection of two sets is the empty set.
Eg: IfA = {1,3,5,7,9} and B = {2,4,6,8,10},
thenAB = , S
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3
5
79
2
46
8
10
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Set Operation
Set Difference
The difference of set A to B denoted as A-B is setof those elements that are In the set B
i.e: A-B={x:x A and xB}
Similarly B-A={x:x B and xA}
A general A-B B-A
Eg:
If A={a,b,c,d} and B={b,c,d,f} then A-B={a,d} and
B-A={e,f}
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Set Operation
Set Complimentary
The complementary event A is the set of allelements which do not belong to it. It is often
symbolized by A or A. All sampling points of apopulation are either in A or in A'. And no sample
point can be a member of both A and A.
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Set Operation
Characteristics of set
Identity law A U = A
A = ADomination law A U = U
A = Idempotent laws A A = A
A A = A
Complementation law (A) = A
Commutative laws A B = B AA B = B A
Associative laws A (B C) = (A B) CA (B C) = (A B) C
Distributive laws A (B C) = (A B) (A C)A (B C) = (A B) (A C)
De Morgans laws A B = A BA B = A B
Absorption laws A (A B) = AA (A B) = A
Complement laws A A = UA A =
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Generalised Union
Generalised Intersection
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Example Generalised Union and Intersection
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Cartesian Product
A new set can be constructed by associating every element of one set with
every element of another set. The Cartesian productof two sets A and B,denoted by A B is the set of all ordered pairs (a, b) such that a is a
member ofA and b is a member ofB.
Examples:
What is the Cartesian product of A ={1, 2} and B ={a, b, c}?Solution: The Cartesian product A B is
A B ={(1, a), (1, b), (1, c), (2, a), (2, b), (2,c)}.
Note that the Cartesian products A B and B A are not equal, unless A =or B = (so that A B =)or A = B
Some basic properties of Cartesian products:
A = . A (BC) = (AB) (AC). (AB) C= (AC) (BC).