CHAPTER 2 : CONCEPT OF SETS.
-->Way of
listing the elements of Sets
There are two ways of writing sets:
1. Roster Method
-listing the elements in any order and enclosing them with
braces.
Example:
A= {January, February, March…December}
B={1,3,5…}
Sometimes the roster
method is used to describe a set without listing all its members. Some
members of the set
are listed, and then ellipses (. . .) are used when the general
pattern of the
elements
is obvious.
Example:
C
= {1,2,3,4,5...100}
2.
Set Builder notation
-characterize all
those elements in the set by stating the property or properties they must have
to be members.
Example:
- as in {x : (x is prime) ∧ (x < 10)}.
-->Specifying
properties of sets
-->
Properties
of Sets
Definition:
~A set is an unordered collection of objects,
called elements or members of the set. A set is said to contain its elements.We
write a ∈
A to denote that a is an element of the set A. The notation a
∈ A denotes that a is not an element of the set A.
∈ A denotes that a is not an element of the set A.
~For example, the notation {a, b, c, d}
represents the set with the four elements a, b, c, and d. This way of
describing a set is known as the roster method.
EXAMPLE
1
The set V of all vowels in the English alphabet can be written as V = {a, e, i, o, u}.
EXAMPLE
2
The set O of odd positive integers less than 10 can be expressed by O = {1, 3, 5, 7, 9}.
EXAMPLE
3
The set of positive integers less than 100 can be denoted by {1, 2, 3, . . . , 99}.
Another way to describe a set is to use set builder notation. We characterize
all those elements in the set by stating
the property or properties they must have to be members. For instance, the
set O of all odd positive integers less than 10 can be written as
O = {x | x is an odd positive integer less
than 10},
or, specifying the universe as the set of
positive integers, as
O = {x ∈ Z+ | x is odd and x < 10}.
We often use this type of notation to
describe sets when it is impossible to list all the elements
of the set. For instance, the set Q+ of all
positive rational numbers can be written as
Q+ = {x ∈ R | x = p
q , for some positive integers p and q}.
Beware!
that mathematicians
Disagree whether 0 is a natural number.We
consider it quite natural.
These sets, each denoted using a boldface
letter, play an important role in discrete mathematics:
N = {0, 1, 2, 3, . . .}, the set of natural
numbers
Z = {. . . ,−2,−1, 0, 1, 2, . . .}, the set
of integers
Z+ = {1, 2, 3, . . .}, the set of positive
integers
Q = {p/q | p ∈ Z, q ∈ Z, and q
= 0}, the set of rational numbers
= 0}, the set of rational numbers
R, the set of real numbers
R+, the set of positive real numbers
C, the set of complex numbers.
Set
Membership
1. An element, or member, of a set is any
one of the distinct objects that make up that set.
2. Writing A = {1, 2, 3, 4 } means that the
elements of the set A are the numbers 1, 2, 3 and 4.
3. Sets of elements of A, for example {1,
2}, are subsets of A.
4. Sets can themselves be elements. For
example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3,
and 4. Rather, there are only three elements of B, namely the numbers 1 and 2,
and the set {3, 4}.
5. The elements of a set can be anything.
For example, C = { apple, orange, plum }, is the set whose elements are the
fruits apple, orange and plum.
Notation
and terminology
1. The relation "is an element
of", also called set membership, is denoted by ∈.
2. Writing x ∈
A means that "x is an element of A".
3. Equivalent expressions to mean set
membership are :
a) x is a member of A
b) x belongs to A
c) x is in A
d) x lies in A
e) A includes x
f) A contains x
g) x is a subset of A
The negation of set membership is denoted
by ∉.
-->Empty
set
-->
EMPTY SET
A special set that
has no element. This called ‘empty set’ or ‘null set’ and it is denoted by ∅.
The empty set also denoted by {}.
Example
The other name of the
empty set is null set Ļ. Consider two sets I = {a, b, c, d}
and J = {1, 2, 3, 4, 5, 6}. Consider another set K which represents the
intersection of X and Y. There is no common element of the set X and Y. So,
intersection of X and Y is null.
K = { }
Cardinality of Empty
Set
The cardinal
number represents the number of elements that are present in the set.
As we know
there are no element in an empty set. So, the cardinal number of empty set is zero.
Properties of Preparation for Empty Set
- Empty set is considered as
subset of all sets. Ļ⊂X
- Union of empty set Ļ with
a set X is X. A∪Ļ=A
- Intersection of an empty set
with a set X is an empty set.
Sets Of Numbers
1. Natural numbers
The whole
numbers starting at 1, are called the natural numbers.
This set is sometimes denoted by N. So N = {1, 2, 3, ...}
When we
write this set by hand, we can't write in bold type so we write an N in blackboard bold font:
2. Integers
All whole numbers, positive, negative and zero form the set of integers. It is sometimes denoted by Z. So Z = {..., -3,
-2, -1, 0, 1, 2, 3, ...}
In
blackboard bold, it looks like this:
3. Real Numbers
The set of real numbers is sometimes denoted
by R.
A real
number may have a finite number and an infinite number of
decimal digits. Example of a finite number is 3.625 and for infinite number is
:
- recur; e.g.
8.137137137...
- ... or they
may not recur; e.g. 3.121597653...
In
blackboard bold:
4. Rational Numbers
The set of rationals is sometimes denoted by
the letter Q.
A rational
number can always be written as exact fraction x/y; where x and y are integers. If x equals 1, the fraction is just the integer. Note that y may NOT equal zero as the value is then undefined.
- For
example: 0.5, -17, 2/17, 82.01, 3.282828... are all rational numbers.
In
blackboard bold:
5. Irrational Numbers
If a number can't be represented exactly by a fraction p/q, it is said to be irrational.
For examples
include: √2, √3, Ļ.
SETS EQUALITIES
Definition :
A set is a collection of things called
the elements, or members.
~ To say x
is the element of G, we write
x ∈ G
How to say it?
# “x belongs
to G”, “G contains x”, “x is in G”.
~ To say x
is not the element of G, we write
x ∈ G
~ To say x and y belong to G, we write
x, y ∈ G
~ To say no elements in a set, we write Ļ
Reminder!
A set with exactly one
element is called a singleton!!
Equality of sets, notation for sets
~ Two sets are identical
(equal) if they contain the same elements.
Example :
S = {a, b, c}
T = {c, b, a}
Notice that {a, b, c} = {c, b, a}
The elements are the same but in
different arrangement.
S
and T are identical sets.
~ Sets may have sets as their elements.
Example :
F = {∅}
← 1 element.
H = {{∅}, ∅}
← 2 elements. H has F as 1 of its elements.
~ Sets can be finite or infinite.
Example :
Finite set : {0, 1, 2, 3, 4, 5, 6, 7, 8,
9, . . . , 20}
Infinite set : {. . . , 7, 8, 9, . . . }
~ Let’s say N is a set of natural numbers and Z is a set of integers.
#
Use set construction notation.
Example :
a) {n | n ∈ N and n is even} ← is the set of even natural numbers.
b) {n | n ∈ Z and 3 | n} ← the integers that are multiples of 3.
VENN DIAGRAMS
Ć Represented the sets graphically
Ć Universal set (U), contains all the
objects under consideration and is represented by a rectangle
Ć Inside rectangle are other
geometrical figures that used to represent sets
The intersection of
sets A and B is those elements which are in set A and set
B. A diagram showing the intersection of A and B is on the left.
The union of sets
A and B is those elements which are in set A or set
B or both. A diagram showing the union of A and B is on
the right.
Subsets :
Subsets :
Definition:
Set B is a subset of a set A if and only if every object of
B is also an object of A.
We write B Ć
A
Proper subset:
Set B is a proper subset of set A, if there exists an element in A that does not belong to B. we write B Ć A Having said that, B is a proper subset of A because f is in A, but not in B. We write B Ć A instead of B Ć A Universal set: The set that contains all elements being discussed In our example, U, made with a big rectangle, is the universal set Set A is not a proper subset of U because all elements of U are in subset A Notice that B can still be a subset of A even if the circle used to represent set B was not inside the circle used to represent A. This is illustrated below:  As you can see, B is still a subset of A because all its objects or elements (c, and d) are also objects or elements of A. B is again a proper subset because there are elements of A that does not belong to B A and B are also subsets of the universal set U, but especially proper subsets since there are elements in U that does not belong to A and B In general, it is better to represent the figure above as show below to avoid being redundant: 
The area where elements c, and d are located is the
intersection of A and B.
Power
Set
POWER SET
Definition :
A power set is a set of which the elements are all the
subsets of a given set S including the empty set.
Notation:
The number of members of a set is often written as |S|, so
we can write :
|P(S)|
= 2n
Properties:
·
If X is finite, then |P (X)| = 2|X|
·
The above property also holds when X is not
finite. For a set X, let |X| be the cardinality of X . then |P(X)| = 2|X|
=|2X|,where 2X is the set of all function from X to
{0,1}
·
For an arbitrary set X, Cantor’s theorem states : a0 there is no
bijection between X and P(X), and b) the cardinality of P(X) is tgreater than the cardinality of X.
Example:
For the set S = {1,2,3} how many members will the power set
have?
Answer : |P(S)| = 2n
= 23 = 8
Or
P(S) =
{{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
|
Set operation
1.
Set Operation
-Union
Definition (Union): The union of sets A and B, denoted by A U B , is the set defined as
A U B = { x | x ∈ A v x ∈ B }
Example 1: If A = {1, 2, 3} and B = {4, 5} , then A U B = {1, 2, 3, 4, 5} .
Example 2: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A U B = {1, 2, 3, 4, 5} .
Note that elements are not repeated in a set.
A U B = { x | x ∈ A v x ∈ B }
Example 1: If A = {1, 2, 3} and B = {4, 5} , then A U B = {1, 2, 3, 4, 5} .
Example 2: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A U B = {1, 2, 3, 4, 5} .
Note that elements are not repeated in a set.
-Intersection
A ∩ B = { x | x ∈ A V x ∈ B }
Example 3: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A ∩B = {1, 2} .
Example 4: If A = {1, 2, 3} and B = {4, 5} , then A ∩ B = ∅ .
Note that in general A - B B - A
-Disjoint Set
Two sets, A and B, are
disjoint if they have no
element in common.
A ∩ B = Ē¾
-Set Difference
Definition (Difference): The difference of sets A from B , denoted by A - B , is
the set defined as
A - B = { x | x ∈ A x ∈ B }
Example 5: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A - B = {3} .
Example 6: If A = {1, 2, 3} and B = {4, 5} , then A - B = {1, 2, 3} .
A - B = { x | x ∈ A x ∈ B }
Example 5: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then A - B = {3} .
Example 6: If A = {1, 2, 3} and B = {4, 5} , then A - B = {1, 2, 3} .
-Set Complimentary
Definition (Complement): For a set A, the difference U - A , where U is the universe, is called the complement of A and it is denoted by A-.
Thus is the set of everything that is not in A.
-Characteristics of set
Set can interact with other
set by intersection and union
Zero set is called null set
Generalized Unions and Intersection
Because unions and intersections of sets
satisfy associative laws, the sets A∪B ∪C and A∩B ∩C are well deļ¬ned; that is, the meaning of
this notation is unambiguous when A, B, and C are sets. That is, we do not have
to use parentheses to indicate which operation comes ļ¬rst because A∪(B
∪C)= (A∪B)∪C and A∩(B ∩C)= (A∩B)∩C. Note that A∪B
∪C contains those elements that are in at least one
of the sets A, B, and C, and that A∩B ∩C contains those elements that are in all
of A, B, and C.
DEFINITION 6 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.
We use the
notation
A1 ∪A2 ∪···∪An =
n i=1
Ai
to denote the
union of the sets A1,A2,...,An.
DEFINITION 7
Theintersectionofacollectionofsetsisthesetthatcontainsthoseelementsthataremembers
of all the sets in the collection.
We use the
notation
A1 ∩A2 ∩···∩An =
n
i=1
i=1
Ai
to denote the
intersection of the sets A1,A2,...,An.
EXAMPLE 16 For i
= 1,2,..., let Ai ={i,i +1,i+2,...}. Then,
n i=1
Ai =
n i=1
{i,i +1,i+2,...}={
1,2,3,...},
and
n
i=1
i=1
Ai =
n
i=1
i=1
{i,i
+1,i+2,...}={n,n+1,n+2,...}=An.
Cartesian
Product
Cartesian
product : the product of 2 sets.
~ Symbol “X” is used to denote product operation. The Cartesian product
of two sets “A” and “B” is symbolically represented as :
A ×
B
~ We denote
one such pair within a pair of small brackets like :
(a,b)
where a
∈ A and b
∈ B.
Ordered pair
Example :
A = A set of names = {Ekin, Piqa, Fiza, Farah}
B = A set of state names = {Melaka, Kedah, Pahang, Negeri Sembilan}
All possible ordered pairs formed from two sets are :
(Ekin, Melaka), (Piqa, Kedah), (Fiza, Pahang), (Farah, Negeri Sembilan)
Cartesian product
~ The Cartesian product of two sets is defined in terms of ordered pairs.
~ Cartesian product of two non-empty sets “A” and “B”
= set of all ordered pairs of the elements from two sets.
~ The Cartesian product of a non-empty set with an empty set is
equal to empty set.
A × Ļ =
Ļ
~ If the sets are infinite, then the Cartesian
product will be infinite.
~ Example :
A x B = {(x, y) : x ∈ A and y ∈ B}
Note : ∈ is pronounced as “element of”.
~ To emphasize two ways validity of the conditional statements as in the
case of other set operators :
If (x, y) ∈ A × B ⇔ x ∈ A and y ∈ B
Graphical
representation
Let’s
say set F = {001, 002, 003}
E = {N, D, H}
Ordered pairs : {(N, 001), (N, 002),
(N, 003), (D, 001), (D, 002), (D, 003) , (H, 001), (H, 002), (H, 003)}
~ The elements of one set are
represented as rows, whereas elements of other set are represented as columns.
Example :
# Problem 1 : If (x2−1, y+2) = (0,2)
, find “x” and “y”.
Solution : Two ordered pairs are
equal. It means that corresponding elements of the ordered pairs are equal.
Hence,
For x,
→ x2−1 = 0
x= 1 or −1
and for y,
→ y+2 = 2
y= 0
# Problem 2 : If A =
{5,6,7,2}, B = {3,5,6,1} and C = {4,1,8}, then find (A∩B)×(B∩C) .
Solution : Find out the intersections given in the
brackets.
A = {5,6,7,2} B = {3,5,6,1}
→ A∩B={5,6}
B = {3,5,6,1} C = {4,1,8}
→ B∩C={1}
Thus,
(A∩B)×(B∩C)={(6,1),(5,1)}
Numbers of elements
n (A × B) = pq
p = number of elements in A
q = number of elements in B
Multiple products
~ If A1, A2, ……, An is a finite family of sets, then their Cartesian
product, one after another, is symbolically represented as :
A1 × A2
×……………. × An
This product is set of group of ordered elements. Each group of
ordered elements comprises of “n” elements. This is stated as :
A1 × A2 × … × An = {(x1, x2, …, xn) : x1 ∈ A1, x2 ∈A2, …, xn ∈ An}
Ordered triplets
~ Defined as
:
A × A × A = {(x, y, z) : x, y, z ∈ A}
An = A × A × …… × A, where
“n” is the Cartesian power. If n = 2, then
A2 = A × A
~ Also
called Cartesian square.
~ Example : If A =
{-1, 1}, then find Cartesian cube of
set A.
Solution : Following the method of writing ordered sequence of numbers, the product
can be written as :
A × A × A ={(−1, −1,
−1),(−1, −1, 1),(−1, 1, −1),(−1, 1, 1),(1, −1,−1),
(1, −1, 1), (1,
1, −1), (1, 1, 1)}
The total
numbers of elements are 2 x 2 x2 = 8.
Cartesian Coordinate System
~ Consisted
of ordered triplets of real numbers, represents
Cartesian three dimensional space.
R × R × R = {(x, y, z) : x, y, z ∈ R}
Each of the elements in the ordered triplet is a coordinate along an axis and each
ordered triplet denotes a point in three
dimensional coordinate space.
Cartesian coordinate system
 |
Figure 2: The coordinate of a point is an ordered triplet.
|
Similarly,
the Cartesian product "R ×
R" consisting of ordered
pairs defines a Cartesian plane or Cartesian coordinates of two dimensions. It
is for this reason that we call three dimensional rectangular coordinate system
as Cartesian coordinate system.
Commutative property of Cartesian product
~ The order of elements in the ordered pair depends on the position of sets across product sign.
~ If sets
"A" and "B" are unequal and non-empty sets, then :
A × B ≠ B × A
~ If A × B = B × A, then, A=B.
~ The two way conditional statements can be
symbolically represented with the help of two
ways arrow,
A × B = B × A ⇔ A
= B
Distributive property of product
operator
~ The
distributive property of product operator holds for other set operators like
union, intersection and difference operators.
A × (B∪C) =
(A×B) ∪ (A×C)
A × (B∩C) = (A×B) ∩ (A×C)
A × (B−C) = (A×B) − (A×C)
Where sets “A”,”B” and “C” are non-empty sets.
~ Distributive
property should hold for product operator over
three named operators.
Example :
A = {a, b}, B = {1, 2} and C = {2, 3}
1: For distribution over union operator
⇒LHS = A × (B∪C)
= {a,
b} × {1, 2, 3}
= {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
Similarly,
⇒RHS = (A × B) ∪ (A × C)
= {(a,
1),(a, 2), (b, 1), (b, 2)}∪{(a, 2), (a, 3), (b, 2), (b, 3)}
= {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
Hence,
⇒A × (B∪C) = (A × B)∪(A × C)
2: For distribution over intersection
operator
⇒LHS = A × (B∩C) = {a, b} × {2}
={(a, 2), (b, 2)}
Similarly,
⇒RHS = (A × B)∩(A × C)
= {(a,
1), (a, 2), (b, 1), (b, 2)}∩{(a, 2), (a, 3), (b, 2), (b, 3)}
={(a, 2), (b, 2)}
Hence,
⇒A × (B∩C) = (A ×
B)∩(A × C)
3: For distribution over difference
operator
⇒LHS = A × (B − C) = {a, b} × {1}
={(a, 1), (b, 1)}
Similarly,
⇒RHS = (A × B) − (A × C)
= {(a,
1), (a, 2), (b, 1), (b, 2)} − {(a, 2),
(a, 3), (b, 2), (b, 3)}
={(a, 1), (b, 1)}
Hence,
⇒A × (B − C) = (A × B) − (A × C)
Analytical proof
~ Consider
an arbitrary ordered pair (x, y), which belongs to Cartesian product set “A × (B∪C)”. Then,
→ (x, y) ∈ A × (B∪C)
By the
definition of product of two sets,
→ x ∈ A and y ∈
(B∪C)
By the
definition of union of two sets,
→ x ∈ A and (y ∈ B or y ∈ C)
→ (x ∈ A and y ∈ B) o r(x ∈
A
and y ∈ C)
→ (x, y ) ∈ A × B or (x, y) ∈ A × C
By the
definition of union of two sets,
→ (x, y) ∈ (A × B) ∪ (A×C)
But, we had
started with "A × (B∪C)" and used definitions to show
that ordered pair “(x, y)” belongs to another set. It means that the other set
consists of the elements of the first set – at the least. Thus,
→ A × (B∪C) ⊂ (A × B)∪(A × C)
Similarly,
we can start with "(A × B)∪(A × C)" and
reach the conclusion that :
→ (A × B)∪(A × C)⊂A × (B∪C)
If sets are
subsets of each other, then they are equal. Hence,
→ A × (B∪C) = (A ×
B) ∪ (A × C)
~ Using the
same method above, we can also prove
distribution of product operator over intersection and difference operators.
A × (B∩C) = (A × B)∩(A × C)
A × (B −C) = (A × B)−(A × C)
No comments:
Post a Comment