University of Aberdeen, Computing Science CS1512 Discrete Methods Kees van Deemter

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University of Aberdeen, Computing
ScienceCS1512Discrete MethodsKees van Deemter

• Slides adapted from Michael P. Franks Course
  Based on the TextDiscrete Mathematics Its
  Applications (5th Edition)by Kenneth H. Rosen

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Part 3 Sets

• Rosen 5th ed., 1.6-1.7
• 43 slides, 2 lectures

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Introduction to Set Theory (1.6)

• A set represents an unordered collection of zero
  or more distinct objects.
• Sets are ubiquitous in software.
• All of mathematics can be defined in terms of
  some form of set theory (using predicate logic).

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Intuition behind sets

• Almost anything you can do with individual
  objects, you can also do with sets of objects.
  E.g., you can
• refer to them, compare them,
• In addition, you can also
• check whether one set is contained in another (?)
• determine how many elements it has (?)
• quantify over its elements (using it as u.d. for
  ?,?)

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Basic notations for sets

• For sets, well use variables S, T, U,
• We can denote a set S in writing by listing all
  of its elements in curly braces
• a, b, c is the set of whatever 3 objects are
  denoted by a, b, c.
• Set builder notation For any proposition P(x)
  over any universe of discourse, xP(x) is the
  set of all x such that P(x).

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Basic properties of sets

• Sets are inherently unordered
• No matter what objects a, b, and c denote, a,
  b, c a, c, b b, a, c
• Multiple listings make no difference
• a, a, b, a, b, c, c, c, c a,b,c

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• ( There exists a different mathematical
  construct, called bag or multiset, where this
  assumption does not hold. Using square brackets
  to denote a bag
• a, b, c ? a,a,a,c
• Notation if B is a bag then countB(e)number of
  occurrences of e in B )

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Basic notations

• x?S (x is in S) is the proposition that object
  x is an ?lement or member of set S.
• e.g. 3?N, a?x x is a letter of the alphabet
• Notation x?S ?def ?(x?S)

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Definition of Set Equality

• Consequently, two sets are equal if and only if
  they contain the same elementsST ?def ?x(x?S
  ? x?T)
• It does not matter how the set is defined
• For example 1, 2, 3, 4 x x is an
  integer where xgt0 and xlt5 x x is a
  positive integer whose square
  is gt0 and lt25

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Infinite Sets

• Sets may be infinite (i.e., the number of
  elements is larger than 0, larger than 1, larger
  than 2, etc.)
• Symbols for some special infinite setsN 0,
  1, 2, The Natural numbers.Z , -2, -1,
  0, 1, 2, The integers.R The Real
  numbers, such as 374.1828471929498181917281943125
  
• Blackboard Bold or double-struck font (N,Z,R)
  is also often used for these special number sets.

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A set can be empty

• Suppose we call a set S empty iff it has no
  elements ??x(x?S).
• Prove that ?xy((empty(x) ? empty(y) ? xy)
• Note this formula quantifies over sets!

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Theres only one empty set

• Prove that ?xy((empty(x) ?empty(y)) ? xy)
• Proof by Reductio ad Absurdum
• Suppose there existed a and b such that empty(a)
  and empty(b).
• Thus, ??x(x?a) ? ??x(x?b)
• Suppose a?b. This would mean that either ?x(x?a
  ? ?x?b) or ?x(x?b ? ?x?a)
• But the first case cannot hold, for ??x(x?a).
  The second case cannot hold, for ??x(x?b)
• Contradiction, so QED

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The Empty Set

• We have seen that there exists exactly one empty
  set, so we can give it a name
• ? (the empty set) is the unique set that
  contains no elements whatsoever.
• ? xx?x ... xFalse
• Any set containing exactly one element is called
  a singleton

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Subset and Superset Relations

• S?T (S is a subset of T) means that every
  element of S is also an element of T.
• S?T ?def ?x (x?S ? x?T)
• What do you think about these?
• ??S ?
• S?S ?

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Subset and Superset Relations

• S?T (S is a subset of T) means that every
  element of S is also an element of T.
• S?T ?def ?x (x?S ? x?T)
• What do you think about these?
• ??S ? Yes
• S?S ? Yes

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Subset and Superset Relations

• More notation
• S?T (S is a superset of T) ?def T?S.
• Note ST ? S?T? S?T.
• ?def ?(S?T), i.e. ?x(x?S ? x?T)

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Proper (Strict) Subsets Supersets

• S?T (S is a proper subset of T) means that S?T
  but .
• Example1,2 ? 1,2,3
• We have 1,2,3 ? 1,2,3,
• but not 1,2,3 ?
  1,2,3

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Sets Are Objects, Too!

• The elements of a set may themselves be sets.
• E.g. let Sx x ? 1,2,3then S

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Sets Are Objects, Too!

• The objects that are elements of a set may
  themselves be sets.
• E.g. let Sx x ? 1,2,3then S?,
  1, 2, 3, 1,2, 1,3,
  2,3, 1,2,3
• Note that 1 ? 1 ? 1

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Cardinality and Finiteness

• S (read the cardinality of S) is a measure of
  how many different elements S has.
• E.g., ?0, 1,2,3 3, a,b 2,
  1,2,3,4,5 ____
• If S?N, then we say S is finite.Otherwise, we
  say S is infinite.

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The Power Set Operation

• The power set P(S) of a set S is the set of all
  subsets of S. P(S) x x?S.
• E.g. P(a,b) ?, a, b, a,b.
• Sometimes P(S) is written 2S.Note that
  (certainly for finite S), P(S) 2S.
• It turns out ?SP(S)gtS, e.g. P(N) gt
  N.There are different sizes of infinite sets!

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Review Set Notations So Far

• Set enumeration a, b, c
• and set-builder xP(x).
• ? relation, and the empty set ?.
• Set relations , ?, ?, ?, ?, ?, etc.
• Venn diagrams.
• Cardinality S and infinite sets N, Z, R.
• Power sets P(S).

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Axiomatic set theory

• Various axioms, e.g., saying that the union of
  two sets is also a set
• One key axiom Given a Predicate P, construct a
  set. The set consists of all those elements x
  such that P(x) is true.
• But, the resulting theory turns out to be
  logically inconsistent!
• This means, there exist naïve set theory
  propositions p such that you can prove that both
  p and ?p follow logically from the axioms of the
  theory!
• ? The conjunction of the axioms is a
  contradiction!
• This theory is fundamentally uninteresting,
  because any possible statement in it can be (very
  trivially) proved by contradiction!

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This version of Set Theory is inconsistent

• Russells paradox
• Consider the set that corresponds with the
  predicate x ? x
• S x x?x .
• Now ask is S?S?

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Russells paradox

• Let S x x?x . Is S?S?
• If S?S, then S is one of those objects x for
  which x?x. In other words, S?SBy Reductio, we
  have S?S
• If S?S, then S is not one of those objects x for
  which x?x. In other words, S?SBy Reductio, we
  have S?S
• We conclude that both S?S nor S?S
• Paradox! (Theres no assumption that we can
  blame, so we cannot Reductio again)

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• To avoid inconsistency, set theory must somehow
  change

Bertrand Russell1872-1970
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( One example of sophisticated set theory

• Given a set S and a predicate P, construct a new
  set, consisting of those elements x of S such
  that P(x) is true.
• We will not worry about the possibility of
  logical inconsistency Just be sensible when
  constructing sets. )

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Ordered n-tuples

• These are like sets, except that duplicates
  matter, and the order makes a difference.
• For n?N, an ordered n-tuple or a sequence of
  length n is written (a1, a2, , an). Its first
  element is a1, etc.
• Note that (1, 2) ? (2, 1) ? (2, 1, 1).
• Empty sequence, singlets, pairs, triples, ,
  n-tuples.

Contrast withsets
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• n-tuples have many applications. For example,
• Mathematical structures are often described in a
  fixed order, so you know which element plays
  which role. E.g., (N,lt) is the structure whose
  domain is N, which is ordered by the relation lt

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• Relations are often spelled out by means of
  n-tuples. E.g., here are two 2-place relations
• lt (0,1), (1,2), (0,2), )
• Like-to-watch (John,news),(Mary,soap),(Ellen,m
  ovies)
• The first and second argument of a relation may
  come from different sets, e.g. first element of
  the set of persons
• second element of the set of TV-programs

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Cartesian Products of Sets

• For sets A, B, their Cartesian productA?B ?
  (a, b) a?A ? b?B .
• E.g. a,b?1,2 (a,1),(a,2),(b,1),(b,2)
• John,Mary,EllenxNews,Soap

René Descartes (1596-1650)
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Cartesian Products of Sets

• For sets A, B, their Cartesian productA?B ?
  (a, b) a?A ? b?B .
• E.g. a,b?1,2 (a,1),(a,2),(b,1),(b,2)
• John,Mary,EllenxNews,Soap(John,News),(Mary,
  News),(Ellen,News), (John,Soap),(Mary,Soap),(El
  len,Soap)

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Cartesian Products of Sets

• Note that for finite A, B, A?B AB.
• Note that the Cartesian product is not
  commutative i.e., ??AB(A?BB?A).
• Notation extends to A1 ? A2 ? ? An

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Review of 1.6

• Sets S, T, U Special sets N, Z, R.
• Set notations a,b,..., xP(x)
• Set relation operators x?S, S?T, S?T, ST, S?T,
  S?T. (These form propositions.)
• Finite vs. infinite sets.
• Set operations S, P(S), S?T.
• Next up 1.5 More set ops ?, ?, ?.

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Start 1.7 The Union Operator

• For sets A, B, their?nion A?B is the set
  containing all elements that are either in A, or
  (?) in B (or, of course, in both).
• Formally, ?A,B A?B x x?A ? x?B.
• Note that A?B is a superset of both A and B (in
  fact, it is the smallest such superset) ?A, B
  (A?B ? A) ? (A?B ? B)

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Union Examples

• a,b,c?2,3 a,b,c,2,3
• 2,3,5?3,5,7 2,3,5,3,5,7 2,3,5,7

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The Intersection Operator

• For sets A, B, their intersection A?B is the set
  containing all elements that are simultaneously
  in A and (?) in B.
• Formally, ?A,B A?Bx x?A ? x?B.
• Note that A?B is a subset of both A and B (in
  fact it is the largest such subset) ?A, B
  (A?B ? A) ? (A?B ? B)

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Intersection Examples

• a,b,c?2,3 ___
• 2,4,6?3,4,5 ______

?
4
Think The intersection of University Ave. and W
13th St. is just that part of the road surface
that lies on both streets.
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Disjointness

• Two sets A, B are calleddisjoint (i.e., not
  joined)iff their intersection isempty. (A?B?)
• Example the set of evenintegers is disjoint
  withthe set of odd integers.

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Inclusion-Exclusion Principle

• How many elements are in A?B?Can you think of a
  general formula?(Express in terms of A and
  B andwhatever else you need.)

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Inclusion-Exclusion Principle

• How many elements are in A?B? A?B A ? B
  ? A?B
• Example How many students are on our class email
  list? Consider set E ? I ? M, I s s turned
  in an information sheetM s s sent the TAs
  their email address
• Some students may have done both! E I?M
  I ? M ? I?M

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Set Difference

• For sets A, B, the difference of A and B, written
  A?B, is the set of all elements that are in A but
  not B. Formally A ? B ? ?x ? x?A ? x?B?
  
• Also called The complement of B with respect to
  A.

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Set Difference Examples

• 1,2,3,4,5,6 ? 2,3,5,7,9,11
  ___________
• Z ? N ? , -1, 0, 1, 2, ? 0, 1,
  x x is an integer but not a nat.
  x x is a negative integer
  , -3, -2, -1

1,4,6
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Set Difference - Venn Diagram

• A-B is whats left after Btakes a bite out of A

Set A
Set B
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Set Complements

• The universe of discourse can itself be
  considered a set, call it U.
• When the context clearly defines U, we say that
  for any set A?U, the complement of A, written
  , is the complement of A w.r.t. U, i.e., it is
  U?A.
• E.g., If UN,

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Set Identities

• A??

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Set Identities

• A?? A
• A?U

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Set Identities

• A?? A
• A?U A

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Set Identities

• A?? A A?U
• A?U UA?? ?
• A?A A A?A
• A?B B?A A?B B?A
• A?(B?C)(A?B)?C A?(B?C)(A?B)?C

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Have you seen similar patterns before?
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Read ? ?, ? ?, ?F, UT

• A?? A A?U
• A?U U , A?? ?
• A?A A A?A
• A?B B?A , A?B B?A
• A?(B?C)(A?B)?C ,A?(B?C)(A?B)?C

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Set Identities (dont worry about their names)

• Identity A?? A A?U
• Domination A?U U , A?? ?
• Idempotent A?A A A?A
• Double complement
• Commutative A?B B?A , A?B B?A
• Associative A?(B?C)(A?B)?C ,
  A?(B?C)(A?B)?C

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DeMorgans Law for Sets

• Exactly analogous to (and provable from)
  DeMorgans Law for propositions.

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Proving Set Identities

• To prove statements about sets, of the form E1
  E2 (where the Es are set expressions), here are
  three useful techniques
• 1. Prove E1 ? E2 and E2 ? E1 separately.
• 2. Use set builder notation logical
  equivalences.
• 3. Use a membership table.

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Method 1 Mutual subsets

• Example Show A?(B?C)(A?B)?(A?C).
• Part 1 Show A?(B?C)?(A?B)?(A?C).
• Assume x?A?(B?C), show x?(A?B)?(A?C).
• We know that x?A, and either x?B or x?C.
• Case 1 x?B. Then x?A?B, so x?(A?B)?(A?C).
• Case 2 x?C. Then x?A?C , so x?(A?B)?(A?C).
• Therefore, x?(A?B)?(A?C).
• Therefore, A?(B?C)?(A?B)?(A?C).
• Part 2 Show (A?B)?(A?C) ? A?(B?C). (analogous)

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Method 1 Mutual subsets

• A variant of this method translate into
  propositional logic, then reason within
  propositional logic, then translate back into set
  theory. E.g.,
• Show A?(B?C)?(A?B)?(A?C).Suppose x?A ? (x?B ?
  x?C). Prove (x?A ? x?B) ? (x?A ? x?C).

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Method 2 Membership Tables

• Just like truth tables for propositional logic.
• Columns for different set expressions.
• Rows for all combinations of memberships in
  constituent sets.
• Use 1 to indicate membership in the derived
  set, 0 for non-membership.
• Prove equivalence with identical columns.

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Membership Table Example

• Prove (A?B)?B A?B.

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Membership Table Exercise

• Prove (A?B)?C (A?C)?(B?C).

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Membership Table Exercise

• Prove (A?B)?C (A?C)?(B?C).

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Membership Table Exercise

• Prove (A?B)?C (A?C)?(B?C).

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Membership Table Exercise

• Prove (A?B)?C (A?C)?(B?C).

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Review of 1.6-1.7

• Sets S, T, U Special sets N, Z, R.
• Set notations a,b,..., xP(x)
• Relations x?S, S?T, S?T, ST, S?T, S?T.
• Operations S, P(S), ?, ?, ?, ?,
• Set equality proof techniques

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Representing Sets with Bit Strings

• For an enumerable u.d. U with ordering x1, x2,
  , represent a finite set S?U as the finite bit
  string Bb1b2bn where?i xi?S ? (iltn ? bi1).
• E.g. UN, S2,3,5,7,11, B001101010001.
• In this representation, the set operators?,
  ?, ? are implemented directly by bitwise OR,
  AND, NOT!

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Representing Sets with Bit Strings

• In this representation, the set operators?,
  ?, ? are implemented directly by bitwise OR,
  AND, NOT!
• For example, 2,3,5,7,11 ? 1,3,4,9
• 001101010001 ?
• 010110000100
• 011111010101