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Mathematics Sets and Logic Week 1

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Mathematics Sets and Logic Week 1

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Mathematics Sets and Logic Week 1

1. 1. Chapter 1 :Introduction to Set and LogicSM0013 Mathematics IKhadizah GhazaliLecture 1 – 25/05/2011
2. 2. SM0013 - 24/05/2011 2LEARNING OUTCOMESAt the end of this chapter, you should be ableto :know what set & logic are aboutdefine some basic terminologies in set &logicidentify relations between pairs of setsuse Venn diagrams & the countingformula to solve set equationsbuild & use the truth table
3. 3. SM0013 - 24/05/2011 3Outline (A)1.1 Symbols in set & logic1.2 Set Terminology and Notation1.3 Operations on set1.4 Algebra of sets1.5 Finite Sets and CountingPrinciple.
4. 4. SM0013 - 24/05/2011 4Outline (B)1.6 Statements and Argument1.7 Combining, Conditional andBiconditional Statements1.8 Truth Table1.9 Logic of Equivalent
5. 5. SM0013 - 24/05/2011 5SECTION 1.1Symbols in set & logicSymbols is the central part in set & logic. Here someimportant symbols :
6. 6. SM0013 - 24/05/2011 6
7. 7. SM0013 - 24/05/2011 7∋∋
8. 8. SM0013 - 24/05/2011 8
9. 9. SM0013 - 24/05/2011 9
10. 10. SM0013 - 24/05/2011 10Here is a list of sets that we will referto often, and the symbols are standard.
11. 11. SM0013 - 24/05/2011 11Example 1.1Example 1.1
12. 12. SM0013 - 24/05/2011 12Example 1.2Example 1.2
13. 13. SM0013 - 24/05/2011 13Example 1.3Example 1.3
14. 14. SM0013 - 24/05/2011 14SECTION 1.2Set Terminology & NotationGeorge Cantor (1845-1915), in 1895,was the first to define a set formally.Definition : SetA set is a group of things of the same kindthat belong together.The objects that make up a set arecalledelements or members of the set.
15. 15. SM0013 - 24/05/2011 15Some Properties of SetsThe order in which the elements arepresented in a set is not important.A = {a, e, i, o, u} andB = {e, o, u, a, i} both define the same set.The members of a set can be anything.In a set the same member does not appearmore than once.F = {a, e, i, o, a, u} is incorrect since the element‘a’ repeats.
16. 16. SM0013 - 24/05/2011 16Some NotationConsider the set A = {a, e, i, o, u} thenWe write “‘a’ is a member of ‘A’” as:a ∈ AWe write “‘b’ is not a member of ‘A’” as:b ∉ ANote: b ∉ A ≡ ¬ (b ∈ A)
17. 17. SM0013 - 24/05/2011 17Set representation:There are 4 ways to represent a set.1. Set may be represented by words, forexample:A = the first three natural numbers greaterthan zeroB = the colors red, white, blue, and green
18. 18. SM0013 - 24/05/2011 18Set representation (cont.):2. Another way to represented a set is to listits elements between curly brackets (byenumeration).~ is called the roster method, forexample:C = {1, 2, 3}D = {red, white, blue, green}
19. 19. SM0013 - 24/05/2011 19Set representation (cont.):3. Another kind of set notation commonly used isset-builder notation, for example:The set E of a natural number less than 4 iswritten as
20. 20. SM0013 - 24/05/2011 20Set representation (cont.):4. A set can also be represented by a VanndiagramA pictorial way of representing sets.The universal set is represented by theinterior of a rectangle and the other setsare represented by disks lying within therectangle.E.g. A = {a, e, i, o, u} aeiouA
21. 21. SM0013 - 24/05/2011 21Equality of two SetsA set ‘A’ is equal to a set ‘B’ if and only if bothsets have the same elements. If sets ‘A’ and ‘B’are equal we write: A = B. If sets ‘A’ and ‘B’ arenot equal we write A ≠ B.In other words we can say:A = B ⇔ (∀x, x∈A ⇔ x∈B)E.g.A = {1, 2, 3, 4, 5}, B = {2, 4, 1, 3, 5}, C = {1, 3, 5, 4}D = {x : x ∈ Z ∧ 0 < x < 6}, E = {1, 10/5, , 22, 5}then A = B = D = E and A ≠ C.9
22. 22. SM0013 - 24/05/2011 22Universal Set and Empty SetThe members of all the investigated sets ina particular problem usually belongs tosome fixed large set. That set is called theuniversal set and is usually denoted by ‘U’.The set that has no elements is called theempty set and is denoted by Φ or {}.E.g. {x | x2 = 4 and x is an odd integer} = Φ
23. 23. SM0013 - 24/05/2011 23Cardinality of a SetThe number of elements in a set iscalled the cardinality of a set. Let ‘A’ beany set then its cardinality is denotedby |A| @ n(A)E.g. A = {a, e, i, o, u} then |A| = 5.
24. 24. SM0013 - 24/05/2011 24SubsetsSet ‘A’ is called a subset of set ‘B’ if andonly if every element of set ‘A’ is alsoan element of set ‘B’. We also say that‘A’ is contained in ‘B’ or that ‘B’ contains‘A’. It is denoted by A ⊆ B or B ⊇ A.In other words we can say:(A ⊆ B) ⇔ (∀x, x ∈ A ⇒ x ∈ B)
25. 25. SM0013 - 24/05/2011 25Subset cont.If ‘A’ is not a subset of ‘B’ then it isdenoted by A ⊆ B or B ⊇ AE.g. A = {1, 2, 3, 4, 5} and B = {1, 3} andC = {2, 4, 6} then B ⊆ A and C ⊆ A1 35246BAC
26. 26. SM0013 - 24/05/2011 26Some Properties RegardingSubsetsFor any set ‘A’, Φ ⊆ A ⊆ UFor any set ‘A’, A ⊆ AA ⊆ B ∧ B ⊆ C ⇒ A ⊆ CA = B ⇔ A ⊆ B ∧ B ⊆ A
27. 27. SM0013 - 24/05/2011 27Proper SubsetsNotice that when we say A ⊆ B then itis even possible to be A = B.We say that set ‘A’ is a proper subset ofset ‘B’ if and only if A ⊆ B and A ≠ B.We denote it by A ⊂ B or B ⊃ A.In other words we can say:(A ⊂ B) ⇔ (∀x, x∈A ⇒ x∈B ∧ A≠B)
28. 28. SM0013 - 24/05/2011 28Venn Diagram for a ProperSubsetNote that if A ⊂ B then the Venn diagramdepicting those sets is as follows:If A ⊆ B then the disc showing ‘B’ mayoverlap with the disc showing ‘A’ where inthis case it is actually A = BB A
29. 29. SM0013 - 24/05/2011 29Power SetThe set of all subsets of a set ‘S’ is called thepower set of ‘S’. It is denoted by P(S) or 2S.In other words we can say:P(S) = {x : x ⊆ S}E.g. A = {1, 2, 3} thenP(A) = {Φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}Note that |P(S)| = 2|S|.E.g. |P(A)| = 2|A| = 23 = 8.
30. 30. SM0013 - 24/05/2011 30???Is the empty set, Φ a subset of {3, 5, 7}?If we say that Φ is not a subset of {3, 5, 7},then there must be an element of Φ that doesnot belong to {3, 5, 7}.But that cannot happen because Φ is empty.So Φ is a subset of {3, 5, 7}.In fact, by the same reasoning, the empty setis a subset of every set.
31. 31. SM0013 - 24/05/2011 31SECTION 1.3Operations on SetComplement :The (absolute) complement of a set ‘A’ isthe set of elements which belong to theuniversal set but which do not belong to A.This is denoted by Ac or Ā or Á .In other words we can say:Ac = {x : x∈U ∧ x∉A}
32. 32. SM0013 - 24/05/2011 32Venn Diagram for theComplementAAc
33. 33. SM0013 - 24/05/2011 33Set ComplementationIf U is a universal set and A is a subset of U,then
34. 34. SM0013 - 24/05/2011 34∪nionUnion of two sets ‘A’ and ‘B’ is the set of allelements which belong to either ‘A’ or ‘B’ orboth. This is denoted by A ∪ B.In other words we can say:A ∪ B = {x : x∈A ∨ x∈B}E.g. A = {3, 5, 7}, B = {2, 3, 5}A ∪ B = {3, 5, 7, 2, 3, 5} = {2, 3, 5, 7}
35. 35. SM0013 - 24/05/2011 35Venn Diagram Representation forUnionBAA ∪ B3 572
36. 36. SM0013 - 24/05/2011 36I∩tersectionIntersection of two sets ‘A’ and ‘B’ is the setof all elements which belong to both ‘A’ and‘B’. This is denoted by A ∩ B.In other words we can say:A ∩ B = {x : x∈A ∧ x∈B}E.g. A = {3, 5, 7}, B = {2, 3, 5}A ∩ B = {3, 5}
37. 37. SM0013 - 24/05/2011 37Venn Diagram Representation forIntersectionBAA ∩ B3572
38. 38. SM0013 - 24/05/2011 38DifferenceThe difference or the relative complement of aset ‘B’ with respect to a set ‘A’ is the set ofelements which belong to ‘A’ but which do notbelong to ‘B’. This is denoted by A B.In other words we can say:A B = {x : x∈A ∧ x∉B}E.g. A = {3, 5, 7}, B = {2, 3, 5}A B = {3, 5, 7}{2, 3, 5} = {7}
39. 39. SM0013 - 24/05/2011 39Venn Diagram Representation forDifferenceBAA B3 572
40. 40. SM0013 - 24/05/2011 40Disjoint SetsIf two sets A and B have no elements incommon, that is, ifthen A and B are called disjoint sets.Notice that the circles corresponding to A andB not overlap anywhere because A∩B isempty.A ∩ B = ΦB A
41. 41. SM0013 - 24/05/2011 41Some PropertiesA ⊆ A ∪ B and B ⊆ A ∪ BA∩B ⊆ A and A∩B ⊆ B|A ∪ B| = |A| + |B| - |A∩B|A ⊆ B ⇒ Bc ⊆ AcA B = A ∩ BcIf A ∩ B = Φ then we say ‘A’ and ‘B’ aredisjoint.
42. 42. SM0013 - 24/05/2011 42SECTION 1.4Algebra of SetIn section 1.3 we saw that, given two sets Aand B, the operations union and intersectioncould be used to generate two further setsThese two new sets can then be combinedwith a third set C, associated with the sameuniversal set U as the sets A and B, to formfour further sets∪ ∩A B and A B( ) ( ) ( ) ( )∪ ∪ ∩ ∪ ∪ ∩ ∩ ∩C A B , C A B , C A B , C A B
43. 43. SM0013 - 24/05/2011 43And the compositions of these sets are clearlyindicated by the shaded portion in theVenn diagrams
44. 44. SM0013 - 24/05/2011 44Algebraic Laws on Sets:Sets operations of union, intersection andcomplement satisfy various laws(identities).Let U be the universal set and A, B and Care subsets of U.
45. 45. SM0013 - 24/05/2011 45Algebra of SetsIdempotent lawsA ∪ A = AA ∩ A = AAssociative laws(A ∪ B) ∪ C = A ∪ (B ∪ C)(A ∩ B) ∩ C = A ∩ (B ∩ C)
46. 46. SM0013 - 24/05/2011 46Algebra of Sets cont.Commutative lawsA ∪ B = B ∪ AA ∩ B = B ∩ ADistributive lawsA ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
47. 47. SM0013 - 24/05/2011 47Algebra of Sets cont.Identity lawsA ∪ Φ = AA ∩ U = AA ∪ U = UA ∩ Φ = ΦInvolution laws(Ac)c = A
48. 48. SM0013 - 24/05/2011 48Algebra of Sets cont.Complement lawsA ∪ Ac = UA ∩ Ac = ΦUc = ΦΦc = U
49. 49. SM0013 - 24/05/2011 49Algebra of Sets cont.De Morgan’s laws(A ∪ B)c = Ac ∩ Bc(A ∩ B)c = Ac ∪ BcNote: Compare these De Morgan’s lawswith the De Morgan’s laws that you willfind in logic and see the similarity.
50. 50. SM0013 - 24/05/2011 50ProofsBasically there are two approaches inproving above mentioned laws and anyother set relationshipMathematical NotationUsing Venn diagrams
51. 51. SM0013 - 24/05/2011 51Example 1.4 :Example 1.4 :
52. 52. SM0013 - 24/05/2011 52Solution : a)Method 1;
53. 53. SM0013 - 24/05/2011 53Solution : a)Method 2;
54. 54. SM0013 - 24/05/2011 54Solution :b)
55. 55. SM0013 - 24/05/2011 55SECTION 1.5Finite Sets & Counting PrincipleDefinition : Finite setA set is said to be finite if it containsexactly p elements. Otherwise a set issaid to be infinite.
56. 56. SM0013 - 24/05/2011 56Number of the elements in a set;is determined by simply counting theelements in the set.If A is any set, then n(A) or |A| denotes thenumber of elements in A.
57. 57. SM0013 - 24/05/2011 57Another result that is easily seento be true is the following cases :
58. 58. SM0013 - 24/05/2011 58Case III : Inclusion-ExclusionPrinciple
59. 59. SM0013 - 24/05/2011 59Case IV:
60. 60. SM0013 - 24/05/2011 60Example 1.5 :Example 1.5 :
61. 61. SM0013 - 24/05/2011 61Example 1.6 :Example 1.6 :
62. 62. SM0013 - 24/05/2011 62Example 1.7 :Example 1.7 :
63. 63. SM0013 - 24/05/2011 63