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Set Theory

Set Theory. Section 2.1 Tuesday 20 May. Definition: A set is an unordered collection of objects, called the elements or members of the set. A set is said to contain its elements. The definition does not require any relationship among the members of a set.

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Set Theory

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  1. Set Theory Section 2.1 Tuesday 20 May SETS

  2. Definition: A set is an unordered collection of objects, called the elements or members of the set. A set is said to contain its elements. The definition does not require any relationship among the members of a set. In a set, repeated elements are ignored. Notation: x Sdenotes “x is an element of the set S” x S denotes “xis not an element of the set S” What is a Set? SETS

  3. Enumerate it members, enclosed by curly braces (‘{‘ and ‘}’). Examples: vowels = { ‘a’, ‘e’, ‘i’, ‘o’, ‘u’, ‘y’} weekdays = {“Mon”, “Tue”, “Wed”, “Thu”, “Fri”} N = { 0, 1, 2, … } Setbuilder notation: describe how to “build” the set { exp | predicate } or { expU | predicate } evens = { y | y is even and yN } evens = { yN | y is even } evens = { y | n ( nN y = 2n )} evens = { 2y | yN } Describing a Set SETS

  4. The set of Natural Numbers:N= {0, 1, 2, …} The set of Integers:Z= { …, -2, -1, 0, 1, 2, …} The set of Positive Integers: Z+ = {1, 2, …} The set of Rational Numbers:Q= { p/q | p  Zq  Zq ≠ 0 } The set of Real Numbers: R Some Important Sets SETS

  5. Definition: Two sets are equal if and only if they contain the same elements. Formally, let S and X be sets. Then S = X if and only if the following proposition is true:[x (x S→ x X )][x (x X → x S)] Set Equality SETS

  6. The universal set, denoted U, is the set of all possible elements under consideration. The emptyset, denoted { } or Ø, is the set containing no elements. Note that {Ø} is not the same as Ø. Why not????? Other Special Sets SETS

  7. An informal way to picture a set is using Venn diagrams. Here, we are displaying the set {1,3,5} U • 3 • 5 Venn Diagram SETS

  8. Definition: A set A is a subset of a set B, written A B, if and only if every element of A is also an element of B. Exercise: How would you express this in Pred. Logic? A B  x (xA → x  B) Ais aproper subsetof B, writtenA B, if and only if (A B) (A  B ). Exercise: Express A B as a predicate logic formula. A B  ( x (xA → x  B)  ∃x (x∉A x  B) ) One way to show that A = B is to show that A  B and B  A. That is: A = B (A B) (B  A) Subsets of a Set SETS

  9. The empty set Ø is a subset of all sets: S (Ø  S ) How would you prove this proposition? Every set is a subset of itself: S ( S  S ) How would you prove this proposition? Subsets …. SETS

  10. Definition: Let S be a set. If there are exactly n (distinct) elements in S, we say that S is a finite set, and that n is the cardinality of S, written |S|. vowels = { ‘a’, ‘e’, ‘i’, ‘o’, ‘u’, ‘y’ } is finite and |vowels| = 6. weekdays = {“Mon”, “Tue”, “Wed”, “Thu”, “Fri”} is finite and |weekdays| = 5. Definition:A set is infinite if it is not finite. The set of natural numbers N is infinite The set of real numbers R is infinite Cardinality of a Set SETS

  11. Definition:Given a set S, the power set of S, denotedP(S), is the set of all subsets of S. Example:P({1,2,5}) = {Ø, {1},{2},{5}, {1,2},{1,5},{2,5}, {1,2,5}} Exercise: If |S| = 0, what is|P(S)|? If |S| = 1, what is|P(S)|? If |S| = 2, what is|P(S)|? If |S| = 3, what is|P(S)|? What do you conjecture more generally about |P(S)| ? Power Set of a Set SETS

  12. Definition:An ordered n-tuple, written (a1, a2, …, an), is the ordered collection that has a1 as its first element, a2as its second element,…, and an as its nth element. Two n-tuples are equal if and only if each corresponding pair of their elements is equal. (a1, a2, …, an) = (b1, b2, …, bn)  a1 = b1  a2 = b2  …  an= bn Tuples SETS

  13. Tuples 2-tuples are called (ordered) pairs. (a, b) = (c, d)iff a = c and b = d (a, b)≠(b, a)unless a = b 3-tuples are called (ordered) triples. (a, b, c) = (d, e, f )iff a = d and b = e and c = f (a, b, c)≠(c, b, a)unless a = c 13 SETS

  14. Definition:Let A and B be sets. The Cartesian product of A andB, denoted A × B, is the set of all ordered pairs (a, b) where aA andbB. That is:A × B ={ (a, b)| aAbB} Given:weekdays = {“Mon”, “Tue”, “Wed”, “Thu”, “Fri” }amStartTimes = { “8:00a”, “9:00a”, “10:00a”, “11:00a” } A calendar for scheduling morning meetings: weekdays × amStartTimes IfA ≠ and B ≠ , then(A × B = B × A)  (A=B) How would you prove this? Cartesian Product SETS

  15. Cartesian Product Definition:Let A1, A2, …, An be sets. The Cartesian product of A1, A2, …, An, denoted A1× A2 × … × An, is the set of all ordered n-tuples (a1, a2, …, an ) where aiAi fori = 1, 2, …, n. That is:A1× A2 × …× An={ (a1, a2, …, an )| aiAi fori = 1, …, n} Example: Given: weekdays, amStartTimes, as defined previously, andcseLabs = { PRIP, SENS, ELANS, LINKS, METLAB, DEVOLAB } All possible triples indicating a potential morning start time and day for a meeting of a CSE lab: weekdays ×amStartTimes×cseLabs 15 SETS

  16. Relation Every subset of a Cartesian product is a Relation One possible schedule for the CSE conference room:{(“Mon”, “9:00a”, PRIP), (“Mon”, “11:00a”, SENS), (“Tue”, “10:00a”, GEM), (“Thu”, “10:00a”, GEM), (“Thu”, “11:00a”, ELAN) }weekdays × amStartTimes × cseLabs An example binary relation:{(0,0), (1,1), (2,4), (3,9), (4, 16), …}  N × N What well-know function does this relation describe? sq = { (n, n2) | n N } Chapter 8 is all about relations 16 SETS

  17. Set Operations Section 2.2 SET THEORY

  18. A B U Set Union • Definition: Let A and B be sets. The unionof A and B, denoted A B, is the set containing those elements that are either in A or in B, or in both. • Formally, A B = {x | xA  xB} • Venn Diagram SET THEORY

  19. U A B Set Intersection • Definition: Let A and B be sets. The intersection of A and B, denoted by A ∩B, is the set containing those elements which are in both A and B. • Formally, A ∩B = { x | xAxB } SET THEORY

  20. Set Complement • Definition: Let U be the universal set. The complement of a set S, denoted S,is the set containing elements of U which are not in S. • Formally, S= { x  U | x S } • Venn Diagram: U S S SET THEORY

  21. A Set Difference • Definition:Let A andB be sets. The differenceof A and B, denoted A ‒ B, is the set containing those elements that are in A but are not in B. • A ─Bis the complement of B with respect to A. • A ─B={x | xAx B} U A – B B SET THEORY

  22. Exercise: Let:n range over the natural numbers NA = { 6n | 6n< 41 }B = { 9n | 9n< 41 } Find the following: AB A∩B A – B A∩B A × B SET THEORY

  23. Membership table for set operators • Represent conditions for membership in table form . . . 1= an element in the set 0 = an element is not in the set What similarities do you notice between membership tables and truth tables? SET THEORY

  24. Set Identities • Setstogether withtheoperators { , ∩, ∪} define a special class of Boolean algebraic structures. SET THEORY

  25. Set Identities… SET THEORY

  26. Proving Set Identities – Example. • (A B) = A B • To show this identity we can: • Use the definition thatS = T (S T ) (T  S) (x S)  (x T) • Use Set Builder notation • Use membership tables SET THEORY

  27. Example: Use the definition SET THEORY

  28. Example: Using Set Builder… SET THEORY

  29. Example – Using Membership Table 1= an element in the set 0 = an element is not in the set SET THEORY

  30. ExampleWe can use proven identities to prove new identities. Thus, we have proved that: (A (BC)) = (AB) (AC) SET THEORY

  31. Proof of Set Identities… • We can use Venn diagrams to get some ideas about the correctness of identities. • But, we cannotprove identities using Venn diagrams. SET THEORY

  32. Generalized Union • Definition:The union of a collection (set) of sets is the set containing those elements that are members of at least one set in the collection. • In the case of a finite collection of sets,A0, A1, …, An-1: • In the case of an infinite collection of sets, A0, A1, …:where N is the set of natural number, N = {0, 1, 2, …}. SET THEORY

  33. Example • Let Ai = { i, i + 1, i + 2, ….}. SET THEORY

  34. Generalized Intersection • Definition:Theintersection of a collection (set) of sets is the set containing those elements that are members of all the sets in the collection. • In the case of a finite collection of sets,A0, A1, …, An: • In the case of an infinite collection of sets, A0, A1, …: SET THEORY

  35. Example • Let Ai = { i, i + 1, i + 2, ….}. SET THEORY

  36. Exercise • Let • Find SET THEORY

  37. Disjoint Sets • Definition: Two sets A and B are disjointif their intersection is the empty set, i.e., if A ∩B =Ø. • Definition: The sets A0, A1, …, An-1 are pairwise disjoint if each pair of them is disjoint, i.e., if, for all 0 ≤i, j ≤ n-1, Ai ∩Aj= Ø unless i= j . • Observation: If A0, A1, …, An-1 are finite sets, then the sets are pairwise disjoint and only if • Assume this “observation” is an axiom for now. • To prove it we need a formal definition for the cardinality of a set (see next section). SET THEORY

  38. Principle of inclusion-exclusion 1: Set identities can be be proved using any of the three methods just discussed (exercise). • If A and B are finite sets, then |A B| = |A| + |B| - |A ∩B| Proof: As A  B = (A – B)  (A ∩ B) (B – A)and the sets on the right side are pairwise disjoint, the previous observation implies |A  B| = |A – B| + |A ∩ B|+ |B – A| (*). U A B Moreover, as A = (A – B)  (A ∩ B) and B = (B – A)  (A ∩ B), this same observation implies that |A| = |A – B| + |A ∩ B| (**) and that |B| = |B – A| + |A ∩ B| (***). Solving (**) and (***) for |A – B| and |B – A|, respectively, and substituting them in (*), we get:|A  B| = (|A| – |A ∩ B|) + |A ∩ B|+ (|B| – |A ∩ B|) = |A| + |B| – |A ∩ B|. SET THEORY

  39. Russell’s Paradox • Let S be the set of all sets which do not contain themselves. That is, S = { T | T  T } • Does S contain itself??? SET THEORY

  40. Computer Representation of Finite Sets • Use an arbitrary ordering of the elements of the finite universal set U: (a0, a1, …, an-1) • Represent a subset A of U with the bit string of length n: ith bit is: 1, ifai  A;0, if ai  A • Example:For A = { 6n | n< 40 } and B = { 9n | n< 40 }, we can use strings of length 40:A: (1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,…) B: (1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,…) • What string represents Ø? What string represents U? SET THEORY

  41. Computer Representation… • With the string representation of a set, and the following conventions, we can do any operation • Set complementation: Invert each bit (0 to 1, 1 to 0) • Set union: Do a bit-wise “addition” • 0 + 1 = 1 + 0 = 1 • 1 + 1 = 1, 0 + 0 = 0 • Set Intersection: Do a bit-wise “multiplication” • 0 × 1 = 1 × 0 = 0 • 1 × 1 = 1, 0 × 0 = 0 SET THEORY

  42. Exercise: Using:A: (1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0) B: (1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0) Find the following: AB A∩B A – B A∩B SET THEORY

  43. Note on computer implementations • Some programming languages have the finite set as a standard data type. • The Standard Template Library of C++ has a set class. (The multiset allows repeated elements.) • Real numbers (floats, doubles) typically cannot be members of practical sets because uniqueness ( = test) is not well defined. • Set ops are typically O(log n) or better in execution time, where n is the set size SET THEORY

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