Reviewing Set Theory

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					                                                                                                 Set Theory Basics

                                                                                                 Basic concepts:

                                   Reviewing Set Theory
                                                                                                    Objects, elements, sets
                                                                                                    Enumeration notation: as in f1 2 3g         ;   ;

                  COMP2600 — Formal Methods for Software Engineering
                                                                                                    Is-element-of notation: as in 5 P f1 3 5 7g         ;   ;   ;

                                                                                                 Set equality and notational ambiguity:

                                         Clem Baker-Finch                                           f1 2 3g = f3 2 1g
                                                                                                         ;   ;           ;   ;

                                    Australian National University
                                                                                                    f1 2 3 2 1g = f1 2 3g
                                                                                                         ;   ;   ;   ;           ;   ;

                                         Semester 2, 2009

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         Why should we study set theory?                                                         Standard sets

         Understanding mathematics: It is the foundation!                                        Named sets
         Programming: Types and classes are sets.                                                   numbers:
         Specifying software: Legal inputs and appropriate outputs are sets.                         N is the set of natural numbers; Z is the set of integers;
                                                                                                     Q is the set of rational numbers; R is the set of real numbers.
         Reasoning about programs: This inevitably relies on sets of data values.
                                                                                                    booleans: The set fT Fg is typically called Bool.

         Semantics: The meaning of a program is a mapping of one set of data to
                                                                                                    characters: Often consisting of the set of ASCII characters, and often
                                                                                                      will be referred to as Char.

                                                                                                 The empty set           ?
                                                                                                    Basic properties: V x P D x TP ? and V A P D ? & A
                                                                                                                                             :                       :

                                                                                                    Note: f?g is not the same set as ?.
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         Notation for Infinite Sets                                                      Russell’s Paradox

         The “dot-dot-dot” notation                                                     Reveals a problem with an unconstrained notion of sets: even if   P is a
                                                                                        sensible predicate, it doesn’t necessarily characterise a set.
            f1 3 5 7 g ‘obviously’ denotes the set of odd numbers.
                 ;   ;   ;   ; :::

                                                                                        Russell considered the set of all sets that are not members of themselves:
                                                                                               S  fA j A TP Ag
            Is it adequate?
              Occasionally, but in general, no.                                         and posed the question “Is    S a member of itself?”
            Is it ambiguous?                                                           Consider the 2 cases:
             Is f1 2 3 g the same set as f1 2 3 4 g ?
                     ;   ;   ; :::                ;   ;   ;   ; :::
                                                                                          Suppose S P S . Then S is in the set fA j A TP Ag, so S TP S .
                                                                                          Suppose S TP S . Then S satisfies the predicate A TP A, so S P S .
                                                                                        Hence there is no such S . The predicate is sensible, but it does not
                                                                                        characterise a set.

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         Characteristic Predicates                                                      Universal Sets

         Need a better notation:                                                        Venn diagrams are a notation where union and intersection can be well
                                                                                        illustrated. If complements are to have any meaning then Venn diagrams
           S = fx P D j P (x )g                                                        often show a box that corresponds to the whole domain of discourse — the
           P will be called the characteristic predicate of the set S .                universal set.
         We often write fx j P (x )g when the domain is understood.                                                                            B
         We should ask ourselves “Is this always a sensible notation?”

                                                                                                                            A B

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         New sets from Old                                                                           Cartesian Product

         Subset:                                                                                     Notation:

            Notation: A & B holds iff V x P A x P B       :
                                                                                                        A ¢ B = f(x y ) j x P A ” y P B g

            If P is a predicate of appropriate type, and A is given, then                              A ¢ B ¢ C = f(x y z ) j x P A ” y P B ” z P C g
                                                                                                                                   ;   ;

             fx P A j P (x )g is a new set.                                                             etc.
             Of course it is a subset of A.
                                                                                                     Apparent abuse of notation: The product is defined in terms of a set of
         On subsequent slides we recall other ways to build sets.                                    ordered pairs. Since the set they belong to is the Cartesian product, the
            Power set                                                                               definition seems circular.

            Cartesian Product                                                                       Resolution: The existence of the Cartesian product is an axiom of set theory.
                                                                                                     Consequently the expression           f(x y ) P A ¢ B j x P A ” y P B g
                                                                                                                                              ;                                is well
            Union and intersection                                                                  formed.

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         New sets from Old - II                                                                      The Usual Binary Operations

         Power set:                                                                                  Binary union and intersection (of 2 sets belonging to a universe of
            Notation: P (A) denotes the power set of A.
                                                                                                       A ‘ B = fx P U j (x P A) • (x P B )g
            Definition: P (A)  fs j s & Ag
             i.e. P (A) is the set of all subsets of A.                                                A ’ B = fx P U j (x P A) ” (x P B )g
            Example: P (f1 2g) = f? f1g f2g f1 2gg
                                   ;              ;    ;       ;   ;
                                                                                                     U is the universe of discourse. That is, A & U        and   B & U.
                                                                                                     Iterated union and intersection

         Cardinality notation:         jAj   indicates the cardinality of         A.                    Union: Si 2I Di = fx P U j W i P I x P Di g  :

         The power set is so called because           j P (A) j        =   2jAj                         Intersection: Ti 2I Di = fx P U j V i P I x P Di g:

                                                                                                     I is called the index set.

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         Functions and Relations as Sets                                                               Reference Details

         Set-theoretic view: relations and functions are both sets of ordered pairs.                   Chapter 5 of Grassman and Tremblay is called Sets and Relations.

         Relations:                                                                                   5.1 Sets and Set Operations. You must know this.

            Any subset of A ¢ B is a relation between elements of A and elements                     5.2 Tuples, Sequences and Power-sets.
             of B .                                                                                     5.2.1 Introduction: should know.

            That is, any R P P (A ¢ B ) is a relation between A and B .                                5.2.2 Tuples and Cartesian Products: must know.

            If xRy then (x y ) P R, x P A and y P B .
                                                                                                        5.2.3 Sequences and Strings: We’ll get to this.
                                                                                                        5.2.4 Power-sets: should know.
                                                                                                        5.2.5 Types and Signatures: We’ll cover this.
            Each function f from A to B , is also an element of P (A ¢ B ), but                      5.3 Relations: We’ll be talking about this.
             subject to the condition that if (x y1 ) P f ” (x y2 ) P f then y1 = y2 .
                                               ;              ;

                                                                                                      5.4 Properties of Relations: We’ll be talking about this.

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