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					Module #3 - Sets




              Sections 2.1 & 2.2: Sets &
                   Set Operations




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Module #3 - Sets



                     Introduction
       • A set is an unordered collection of objects
         (elements, members) of the set .




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               Basic notations for sets
       • (1) listing all of its elements in curly braces:
            Examples:
            Finite sets   V= {a, e, i ,o ,u }
                          S= {1, 2, 3,…,99}
            infinite sets:
              N = {0, 1, 2, …} The Natural numbers.
              Z = {…, -2, -1, 0, 1, 2, …} The set of integers.
              Z+ = {1,2,3, …} The set of positive integers.
              R = The set of “Real” numbers


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                   Basic notations for sets
       • (2) Set builder notation: For any
         proposition P(x) over any universe of
         discourse, {x|P(x)} is the set of all x such
         that P(x).
       Example
            “Set of all odd positive integers less than 10”
            O= {x | x is an odd positive integer less than 10}


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                   Basic notations for sets

       • (3) Venn Diagrams                            U: universal set




                                                             John Venn
                                                             1834-1923


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             Definition of Set Equality
       • Two sets are declared to be equal if and
         only if they contain exactly the same
         elements.
       • Example:( order and repition do not
         matter)
          {1 ,3 , 5} = {3, 5, 1} = {1, 3, 3, 3, 5, 5, 5}


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                    Membership in sets
       • x∈S : x is an ∈lement or member of set S.
            – e.g. 3∈N, “a”∈{x | x is a letter of the alphabet}
       • x∉S :≡ ¬(x∈S)               “x is not in S”

            – Set equality can be defined in terms of ∈
              relation:
              S=T ↔ (∀x: x∈S ↔ x∈T)


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                    The Empty Set
       • ∅ (“null”, “the empty set”) is the unique set
         that contains no elements.
       • ∅ = {}

       • Empty set ∅ does not equal the singleton set {∅}
                               ∅ ≠ {∅}



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                    Subset Relation
       • S⊆T (“S is a subset of T”) means that every
          element of S is also an element of T.
       • S⊆T ⇔ ∀x (x∈S → x∈T)
       • ∅⊆S, S⊆S.
        If ( S ⊆ T) is true and ( T ⊆ S) is true then S = T
                        ∀x( x∈S ↔ x∈T)
       • S ⊆ T means ¬(S⊆T), i.e. ∃x(x∈S ∧ x∉T)
             /


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                         Proper Subset
       • S⊂T (“S is a proper subset of T”) means
         every element of S is also an element of T,
         but S ≠ T.                    Example:
                                       {1,2} ⊂{1,2,3}



                          S
                                     T
                   Venn Diagram equivalent of S⊂T


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                   Sets Are Objects, Too!
       • The objects that are elements of a set may
         themselves be sets.
       • E.g. let S={x | x ⊆ {1,2,3}}
         then S={∅,
                   {1}, {2}, {3},
                   {1,2}, {1,3}, {2,3},
                   {1,2,3}}
       • Note that 1 ≠ {1} ≠ {{1}} !!!!
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             Cardinality and Finiteness
       • |S| (read “the cardinality of S”) is a measure
         of how many different elements S has.
       • E.g. |∅|=0
               |{1,2,3}| = 3
               |{a,b}| = 2
               |{{1,2,3},{4,5}}| = 2


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               The Power Set Operation
       • The power set P(S) of a set S is the set of all subsets of S.
       • E.g.
         P( {a, b}) = {∅, {a}, {b}, {a, b}}.
         P( {a}) = {∅,{a})
         P({∅})= {∅,{∅}}
         P(∅) = {∅}

       Note:
       if a set has n elements then P(S) has 2n elements.



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                   Ordered n- tuples
       • These are like sets, except that duplicates
         matter, and the order makes a difference.

       • Note that (1, 2) ≠ (2, 1) ≠ (2, 1, 1).

       • Note: 2- tuples are called ordered pairs


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             Cartesian Products of Sets
       • For sets A, B, their Cartesian product
         A×B :≡ {(a, b) | a∈A ∧ b∈B }.
       • E.g. {a,b}×{1,2} = {(a,1),(a,2),(b,1),(b,2)}
       • Note that for finite A, B, |A×B|=|A||B|.
       • Note that the Cartesian product is not
         commutative: A×B≠B×A.
       • E.g. {1,2}×{a,b} = {(1,a),(1,b),(2,a),(2,b)}

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               Section 1.7:Set Operations
                 The Union Operator ∪
       • For sets A, B, the A∪B is the set containing
         all elements that are either in A, or in B or
         in both.
       • Formally:
              A∪B = {x | x∈A ∨ x∈B}.




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                   Union Example
       {2,3,5}∪{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}




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            The Intersection Operator ∩
       • For sets A, B, their intersection A∩B is the
         set containing all elements that are in both
         A and in B.
       • Formally:
               A∩B={x | x∈A ∧ x∈B}.




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                   Intersection Examples
       • {a,b,c}∩{2,3} = ∅     disjoint
       • {2,4,6}∩{3,4,5} = {4}




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            Inclusion-Exclusion Principle
       • How many elements are in A∪B?
          |A∪B| = |A| + |B| − |A∩B|
       • Example:
       {1,2,3} ∪ {2,3,4,5} = {1,2,3,4,5}
       {1,2,3} ∩ {2,3,4,5} = {2,3}
       | {1,2,3,4,5}| = 3+4-2=5


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                    Set Difference
       • For sets A, B, the difference of A and B,
         written A−B, is the set of all elements that
         are in A but not in B.
       A − B :≡ {x | x∈A ∧ x∉B}
              = {x | ¬(x∈A → x∈B) }
       called: complement of B with respect to A.
       Ex: {1,2,3,4,5,6}-{2,3,5,7,9,11}={1,4,6 }

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            Set Difference - Venn Diagram



                   Set
                   A−B

                    Set A              Set B
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                       Set Complements
       • U: universe of discourse
       • A : For any set A⊆U, the complement of A,
         i.e., it is U−A.
       •
                 A = {x | x ∉ A}               A
                                                              A
       • E.g., If U=N,
                                                U
            {3,5} ={0,1,2,4,6,7,...}

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                     Set Identities
       •    Identity:    A∪∅ = A = A∩U
       •    Domination: A∪U = U , A∩∅ = ∅
       •    Idempotent:   A∪A = A = A∩A
       •    Double complement: ( A ) = A
       •    Commutative: A∪B = B∪A , A∩B = B∩A
       •    Associative: A∪(B∪C)=(A∪B)∪C ,
                         A∩(B∩C)=(A∩B)∩C

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             DeMorgan’s Law for Sets


                   A∪ B = A ∩ B
                   A∩ B = A ∪ B



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                   Proving Set Identities
       To prove statements about sets, of the form
           E1 = E2 (where the Es are set expressions),
           here are three useful techniques:
       1. Prove E1 ⊆ E2 and E2 ⊆ E1 separately.
       2. Use set builder notation &
           logical equivalences.
       3. Using set identities

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             Example: Show A∩B= A∪ B
            method1: Prove E1 ⊆ E2 and E2 ⊆ E1
        • Part 1: Show A∩B ⊆ A ∪ B
        • Assume x∈ A∩B then :
             – x ∉ A∩B by the definition of the complement
             – ~((x ∈ A) ∧ (x ∈ B)) by the definition of intersection
             – ~(x ∈A) ∨ ~ ( x ∈B) by DeMorgan’s law
             – x∉A∨ x∉B                by the definition of negation
             – x ∈ A ∨ x ∈ B by the definition of the complement
             – x∈A∪B             by the definition of union
             This shows that : A∩B ⊆ A ∪ B
        • Part 2: Show A ∪ B ⊆ A ∩B


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            Method2: set builder notation
       • Show A∩B = A ∪ B
       • A∩B= {x | x ∉ A∩B}
             = {x | ~( x ∈ (A∩B))}
             = {x | ~( x ∈ A ∧ x ∈B)}
             = {x | x ∉ A ∨ x ∉ B}
             = {x | x ∈ A ∨ x ∈ B)}
             = {x | x ∈ A ∪ B } = A ∪ B

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A ∪ (B ∩ C)
 Module #3 - Sets



                 Method3 : using set identities
              • Ex: prove that using set identities:
              • A ∪ (B ∩ C)      = ( C ∪ B) ∩ A




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