University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter

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University of Aberdeen, Computing
ScienceCS3511Discrete 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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Module 3The Theory of Sets

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

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Introduction to Set Theory (1.6)

• A set is another type of structure, representing
  an unordered collection of zero or more distinct
  objects.
• Set theory deals with operations between,
  relations among, and statements about sets.
• Sets are ubiquitous in computer software systems.
• All of mathematics can be defined in terms of
  some form of set theory.

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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. (informally speaking), you can
• refer to them, compare them, combine them,
• You can also do some things to a set that you
  probably cannot do to an individual E.g., you
  can
• 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, c, c, c, ca,c.

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Basic properties of sets

• There exists a different mathematical construct,
  called bag or multiset, where this assumption
  does not hold. Using square brackets, we have
• a,a,c,c,c,ca,c,a,c,c,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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Definition of Set Equality

• Two sets are equal if and only if they contain
  exactly the same elements.
• 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., not finite, without
  end, unending).
• 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.
• Infinite sets come in different sizes!

More on this after module 4 (functions).
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Venn/Euler Diagrams
John Venn1834-1923
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0
4
6
8
1
Even integers from 2 to 9
-1
3
5
7
9
Odd integers from 1 to 9
Positive integers less than 10
Primes lt10
Integers from -1 to 9
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• Warning such diagrams come in different flavours
  (e.g., Venn or Euler). We will mix and match
  flavours This is ok as long as its clear what
  we mean.

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Basic Set Relations Member of

• 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
• Set equality is defined in terms of ?ST ?def
  ?x x?S ? x?TTwo sets are equal iff they have
  the same members.
• Notation x?S ?def ?(x?S)

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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.

2
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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, because 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 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,

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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)
• If R is a relation between A and B then R?AxB

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Cartesian Products of Sets

• Note that
• for finite A, B, A?B A.B
• the Cartesian product is not commutative i.e.,
  ??AB A?BB?A.
• notation extends naturally 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?? ?

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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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( An algebraic perspective

• Propositional logic and set theory are
  isomorphic.
• They both instantiate what is known as a Boolean
  Algebra
• A structure (D,?,, . ,0,1) where
• ? is a one-place operation
• and . are a two-place operations
• is commutative, etc.
  )

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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 3 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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Generalized Unions Intersections

• Since union intersection are commutative and
  associative, we can extend them from operating on
  ordered pairs of sets (A,B) to operating on
  sequences of sets (A1,,An), or even on unordered
  sets of sets,XA P(A) (for some property
  P).
• (This is just like using ? when adding up
  large or variable numbers of numbers)

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Generalized Union

• Binary union operator A?B
• n-ary unionA?A2??An ? ((((A1? A2) ?)?
  An)(grouping order is irrelevant)
• Big U notation
• Or for infinite sets of sets

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Generalized Intersection

• Binary intersection operator A?B
• n-ary intersectionA1?A2??An?((((A1?A2)?)?An)
  (grouping order is irrelevant)
• Big Arch notation
• Or for infinite sets of sets

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(Aside Representations

• A frequent theme of this course is methods of
  representing one discrete structure using another
  discrete structure.
• E.g., one can represent natural numbers as
• Sets 0??, 1?0, 2?0,1, 3?0,1,2,
• Can you write 3 more fully?

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Representations

• Sets 0??, 1?0, 2?0,1, 3?0,1,2,
• General n ? x? N xltn
• Can you write 3 more fully?
• 0 ?
• 1 ?
• 2 ?,?
• 3 ?,?,?,?

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Representations

• 3 ?,?,?,?
• Note that this uses ? as the only building block.
  (This is how pure set theory works everything
  is created from nothing ) For Computer Science,
  this is not directly relevant.
  )

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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

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• We now know enough about sets to move on to
  relations between sets, and functions from one
  set to another