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Chapter 1:
Discrete Structures
SETS
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SETS
• WHY ARE WE STUDYING SETS
• The concept of set is basic to all of
mathematics and mathematical applications.
• Sets are foundational in many areas ofComputer Science.
• For us, sets are useful to understand the
principles of counting and probability theory
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SETS
A set is determined by its elements and
not by any particular order in which the
element might be listed.Example,
A={1, 2, 3, 4},
A might just as well be specified as
{2, 3, 4, 1} or {4, 1, 3, 2}
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SETS
The elements making up a set are
assumed to be distinct, we may have
duplicates in our list, only one occurrenceof each element is in the set.
Example
{a, b, c, a, c} = {a, b, c}
{1, 3, 3, 5, 1} = {1, 3, 5}
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SETS
Use uppercase letters A, B, C … to
denote setsThe symbol ∈ stands for “belongs to”
The symbol ∉ stands for “does not
belong to”
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EXAMPLE
X={ a, b, c, d, e },
b∈ X and m∉ X
A={{1}, {2}, 3, 4},
{2}∈ A and 1∉ A
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SETS
If a set is a large finite set or an infinite set,we can describe it by listing a property
necessary for memberships
Let S be a set, the notation, A= {x | x ∈ S, P(x)} or A= {x ∈ S | P(x)}
means that A is the set of all elements x of
S such that x satisfies the property P.
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SETS
Let A={1, 2, 3, 4, 5, 6}, we can also
write A as,
A={x | x ∈ Z, 0<x<7}
if Z denotes the set of integers.
Let B={x | x ∈ Z, x>0},B={1, 2, 3, 4, …}
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SUBSET
If every element of A is an element
of B, we say that A is a subset of B
and write A ⊆ B.
A=B, if A ⊆ B and B ⊆ A.
The empty set { } is a subset of every
set.
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EXAMPLE
A={1, 2, 3}
Subset of A,
{ }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} {1, 2, 3}
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PROPER SUBSET
If A is a subset of B and A is
not equal to B, we say that A
is a proper subset of B.
A⊂ B and A≠B
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EXAMPLE
B={1, 2, 3}
Proper subset of B,
{ }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}
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EQUAL SET
Two set A and B are equal and we write
A=B, if A and B have the same elements.
example A={a, b, c}, B={b, c, a}, A=B
C={1, 2, 3, 4}D={x | x is a positive integer , 2x <10}
C=D
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UNIVERSAL SET
Sometimes we are dealing with sets all
of which are subsets of a set U.
This set U is called a universal set or auniverse.
The set U must be explicitly given or
inferred from the context.
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EXAMPLE
The sets X={1,2,3}, Y={2,4,6,8} and Z={5,7}
One may choose U={1,2,3,4,5,6,7,8} as auniversal set.
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Finite Set
Let A be a set,
If there exists a nonnegative integer
n such that A has n elements, then Ais called a finite set with n elements.
Example
C = {1, 2, 3, 4}
B = {x | x is an integer, 1 < x < 4}
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Infinite SET
Let A be a set, A is called an infinite set, if A is
not a finite set.
Example Z = {x| x is an integer}
or Z = {…, -3, -2, -1, 0, 1, 2, 3,…}
S={x| x is a real number , 1 < x < 4}
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CARDINALITY SET
Let S be a finite set with n distinct
elements, where n≥0.
Then we write |S|=n and say that thecardinality (or the number of elements) of
S is n.
example A= {1, 2, 3}, |A|=3
B= {a,b,c,d,e,f,g}, |B|=7
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POWER SET
The set of all subsets of a set A,
denoted
P(A), is called the power set of A.P(A)= {X | X ⊆ A}
If | A|=n, then |P(A)| = 2^n
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EXAMPLE
A={1,2,3}
the power set of A,
P(A)= {{ }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3},
{1,2,3} }Notice that| A| = 3, and|P(A)| = 2^3 = 8
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EXERCISE
Let X= {1, 2, 2, {1}, a}
Find:
| X|
Proper subset of X
Power set of X
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OPERATION ON SET
The union of two sets A and B, denoted by A
∪ B, is defined to be the set
A ∪ B = { x | x ∈ A or x ∈ B}
The union consists of all elements belongingto either A or B (or both)
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OPERATIONS ON SETS
Venn diagram of A ∪ B
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EXAMPLE
A={1, 2, 3, 4, 5}, B={2, 4, 6} and C={8, 9}
Find : A ∪ B =
A ∪ C =
B ∪ C =
A ∪ B ∪ C =
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A ∪ B = {1, 2, 3, 4, 5, 6}
A ∪ C = {1, 2, 3, 4, 5, 8, 9}
B ∪ C = {2, 4, 6, 8, 9}
A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 8, 9}
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OPERATIONS ON SETS
The intersection of two sets A and B,
denoted by A ∩ B, is defined to be the
set
A ∩ B = { x | x ∈ A and x ∈ B}
The intersection consists of all
elements belonging to both A and B.
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OPERATIONS ON SETS
Venn diagram of A ∩ B
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EXAMPLES
A={1, 2, 3, 4, 5, 6},
B={2, 4, 6, 8, 10} and
C={ 1, 2, 8, 10 }
Find: A ∩ B =
A ∩ C =
C ∩ B = A ∩ B ∩ C =
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A ∩ B = {2, 4, 6}
A ∩ C = {1, 2}
C ∩ B = {2, 8, 10}
A ∩ B ∩ C = {2}
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OPERATIONS ON SETS
Venn diagram, A ∩ B = ∅
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OPERATIONS ON SETS
If A and B are finite sets, the
cardinality of A ∪ B,
| A ∪ B| = |A| + |B| − |A ∩ B|
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OPERATIONS ON SETS
The set
A-B= { x | x ∈ A and x ∉ B}
is called the difference.
The difference A-B consists of all
elements in A that are not in B.
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OPERATIONS ON SETS
Venn diagram of A-B
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EXAMPLE
A= { 1, 2, 3, 4, 5, 6, 7, 8 }
B= { 2, 4, 6, 8 }
A-B = { 1, 3, 5, 7 }
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OPERATIONS ON SETS
The complement of a set A with respect to
a universal set U, denoted by A′ is defined
to be
A′ = { x ∈ U| x ∉ A}
A′ = U-A
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OPERATIONS ON SETS
Venn diagram of A’
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EXAMPLE
Let U be a universal set,
U= { 1, 2, 3, 4, 5, 6, 7 }
A= { 2, 4, 6 }
A′ = U – A = { 1, 3, 5, 7 }
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EXERCISE
Let,
U = { a, b, c, d, e, f , g, h, i, j, k, l, m }
A = { a, c, f, m}
B = { b, c, g, h, m }
Find:
A ∪ B , A ∩ B , | A ∪ B| , A-B and A′.
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SOLUTIONS
A ∪ B = { a,b,c,f,g,h,m }
A ∩ B = { c,m }
| A ∪ B| = 7
A-B = { a,f }
A′ = { b,d,e,g,h,i,j,k,l}
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{1,2,3} ∪ {3,4,5}
{x| x>0} ∪ {x| x>1}
{1,2,3} ∩ {3,4,5
{x| x>0} ∩ {x| x>1}
EXERCISE
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{1,2,3}∩ {3,4,5}={3}
{x| x>0}∩ {x| x>1} = {x| x>1}
{1,2,3}∪ {3,4,5}={1,2,3,4,5}
{x| x>0}∪ {x| x>1} = {x| x>0}
SOLUTIONS
P ti f S t
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Properties of Set
• Commutative laws
• A ∩ B =B ∩ A
• A ∪ B =B ∪ A
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Properties of Set
• Assoc ia tive laws
• A ∩ (B ∩ C ) = (A ∩ B ) ∩ C
• A ∪ (B ∪ C ) = (A ∪ B ) ∪ C
• Distributive laws
• A ∪ (B ∩ C ) = (A ∪ B ) ∩ (A ∪ C )• A ∩ (B ∪ C ) = (A ∩ B ) ∪ (A ∩ C )
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Properties of Set
• De Morgan’s laws
• (A ∩ B )′ = A′ ∪ B ′
• (A ∪ B )′ = A′ ∩ B ′
• Properties of universa l set
• A ∪ U = U • A ∩ U = A
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Properties of SET
• Properties of empty set
• A ∪ ∅ = A
• A ∩ ∅ = ∅
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EXAMPLE
Simplify the set
(((A∪B)∩C)’∪B’)’
SOLUTION
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Cartesian Produc t
• An ordered pa ir (a , b ) is c onsidered
d istinc t from ordered pa ir (b , a ), unless
a =b .
• Examp le (1, 2) ≠ (2, 1)
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Cartesian Produc t
• The Cartesian product of two sets A
and B, written A× B is the set,
• A× B = {(a,b)| a∈ A, b∈ B}• For any set A,
• A×∅ = ∅× A = ∅
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EXAMPLE
• A= {a , b }, B ={1, 2}.
• A×B = {(a , 1), (a , 2), (b , 1), (b , 2)}
• B ×A = {(1, a ), (1, b ), (2, a ), (2, b )}
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Cartesian Produc t
• if A ≠ B, then A× B ≠ B× A.
• if | A| = m and | B| = n, then | A× B|=mn
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EXAMPLE
• A= {1, 3}, B ={2, 4, 6}.
• A×B = {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4),
(3, 6)}
• B ×A = {(2, 1), (2, 3), (4, 1), (4, 3), (6, 1),
(6, 3)}
• A ≠ B , A×B ≠ B ×A
• | A| = 2 , | B | = 3, | A×B | = 2.3= 6.
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EXERCISE
• {1,2 } X {3, 4,5}
• {Male, Female} X {Married, Single} X
{ Student, Faculty}