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Reviewing Set Theory
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 COMP 2600 — Reviewing Set Theory 1 COMP 2600 — Reviewing Set Theory 3 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 another. 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 ?. COMP 2600 — Reviewing Set Theory 2 COMP 2600 — Reviewing Set Theory 4 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: Problems: 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. COMP 2600 — Reviewing Set Theory 5 COMP 2600 — Reviewing Set Theory 7 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. U We often write fx j P (x )g when the domain is understood. B A We should ask ourselves “Is this always a sensible notation?” A B U COMP 2600 — Reviewing Set Theory 6 COMP 2600 — Reviewing Set Theory 8 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. COMP 2600 — Reviewing Set Theory 9 COMP 2600 — Reviewing Set Theory 11 New sets from Old - II The Usual Binary Operations Power set: Binary union and intersection (of 2 sets belonging to a universe of discourse): 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. COMP 2600 — Reviewing Set Theory 10 COMP 2600 — Reviewing Set Theory 12 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. Functions: 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. COMP 2600 — Reviewing Set Theory 13 COMP 2600 — Reviewing Set Theory 14