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  • Graduation Thesis

    0. Mathematical Review: Elementary Set Theory

    Sets and Operators

    The Set is a Mathematical Entity undefinable precisely: we could say that is a collection of elements of various species. We could imagine \(A\) as the set of English Alphabet letters,or the set of the people living on the Earth, or a collection of songs, etc... . We could represent sets in different ways:

    • Describing the property of the elements in braces:

      \(A=\left\lbrace \right. \)The set of the letters of the English alphabet\(\left\rbrace \right. \)

    • Saying the elements one-to-one:

      \(A=\left\lbrace a,b,c,d, \ldots \right\rbrace \)

    • The graphic representation by using the Euler-Venn Diagram:

      image

     Now if e.g. a letter a belongs to the set \(A\) we will write \[a\in A \,\] while if a number e.g. \(1\) does not belong to \(A\) we will write \[1\not\in A.\] The set that does not contain elements is called Empty Set and is represented by the symbol \(\varnothing. \)
    If we consider two sets \(A\) and \(B\) and one of them, \(B\), is included in the other, \(A\), we say that \(B\) is subset of \(A\) and we represents it by writing \[B \subseteq A.\] We define Power Set of \(A\), \(\mathcal{P}(A)\), the set of all subsets of \(A\), including \(A\) and \(\varnothing\). E.g.:

    \(A=\lbrace a,b \rbrace\)  
     
    \(\mathcal{P}(A)=\big\lbrace \varnothing,A,\lbrace a \rbrace, \lbrace b \rbrace \big\rbrace\)

    We note that if \(A\) has \(n\) elements, \(\mathcal{P}(A)\) has \(2^{n}\) elements.
    It’s possible to define some operators over the sets that are: Union, Intersection,Complement and Cartesian Product.

    • The Union represented by the symbol \(\cup\) is the operator that given two sets \(A\) and \(B\) returns the set \(A \cup B\) that contains the elements of both \(A\) and \(B\).
      E.g.:

      \( A=\lbrace a,b\rbrace\)  
      \( B=\lbrace c,d \rbrace \)  
      \(A \cup B =\lbrace a,b,c,d \rbrace\)

    • The Intersection represented by the symbol \(\cap\) is the operator that given two sets \(A\) and \(B\) returns the set \(A \cap B\) that contains the elements that are in common between \(A\) and \(B\).
      E.g.:

      \( \ \ \ \ \ A=\lbrace a,b\rbrace\)  
      \(\ \ \ \ \ \ \ \ \ \ \ B=\lbrace a,b,c,d \rbrace \)  
      \(A \cap B =\lbrace a,b \rbrace\)

    • The Complement represented by the symbol \(\setminus\) is the operator that given two sets \(A\) and \(B\) returns the set \(A \setminus B\) that contains the elements of \(A\) that don’t belong to \(B\).
      E.g.:

      \(\ \ \ \ \ \ \ \ \ \ \ A=\lbrace a,b,c,d \rbrace \)  
      \( \ \ \ \ \ B=\lbrace a,b\rbrace\)  
      \(A \setminus B =\lbrace c,d \rbrace\)

    • The Cartesian Product represented by the symbol \(\times\) is the operator that given two sets \(A\) and \(B\) returns the set \(A \times B\) that is the set of the ordered pairs obtainable taking the first element by \(A\) and the second one by \(B\).

      \( A=\lbrace a,b\rbrace\)  
      \( B=\lbrace c,d \rbrace \)  
      \(A \times B =\lbrace (a,c),(a,d),(b,c),(b,d) \rbrace\)

    Starting by the Cartesian Product we will define the Relations.