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Module #3 - Sets
University of Aberdeen, Computing Science
CS3511
Discrete Methods
Kees van Deemter
Slides adapted from Michael P. Frank‟s
Course Based on the Text
Discrete Mathematics & Its Applications
(5th Edition)
by Kenneth H. Rosen
9/27/2010 Michael P. Frank / Kees van Deemter 1
Module #3 - Sets
Module #3:
The Theory of Sets
Rosen 5th ed., §§1.6-1.7
~43 slides, ~2 lectures
9/27/2010 Michael P. Frank / Kees van Deemter 2
Module #3 - Sets
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.
9/27/2010 Michael P. Frank / Kees van Deemter 3
Module #3 - Sets
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 ,)
9/27/2010 Michael P. Frank / Kees van Deemter 4
Module #3 - Sets
Basic notations for sets
• For sets, we‟ll 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,
{x|P(x)} is the set of all x such that P(x).
9/27/2010 Michael P. Frank / Kees van Deemter 5
Module #3 - Sets
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, c}={a,c}.
9/27/2010 Michael P. Frank / Kees van Deemter 6
Module #3 - Sets
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,c]=[a,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
9/27/2010 Michael P. Frank / Kees van Deemter 7
Module #3 - Sets
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 x>0 and x<5 } =
{x | x is a positive integer whose square
is >0 and <25}
9/27/2010 Michael P. Frank / Kees van Deemter 8
Module #3 - Sets
Infinite Sets
• Sets may be infinite (i.e., not finite, without end,
unending).
• Symbols for some special infinite sets:
N = {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).
9/27/2010 Michael P. Frank / Kees van Deemter 9
Module #3 - Sets
Venn/Euler Diagrams
John Venn
1834-1923
9/27/2010 Michael P. Frank / Kees van Deemter 10
Module #3 - Sets
• Warning: such diagrams come in different
flavours (e.g., Venn or Euler). We will „mix
and match‟ flavours – This is ok as long as
it‟s clear what we mean.
9/27/2010 Michael P. Frank / Kees van Deemter 11
Module #3 - Sets
Basic Set Relations: Member of
• xS (“x is in S”) is the proposition that
object x is an lement or member of set S.
– e.g. 3N, “a”{x | x is a letter of the alphabet}
• Set equality is defined in terms of :
S=T :def x: xS xT
“Two sets are equal iff they have
the same members.”
• Notation: xS :def (xS)
9/27/2010 Michael P. Frank / Kees van Deemter 12
Module #3 - Sets
A set can be empty
• Suppose we call a set S empty
iff it has no elements: x(xS).
• Prove that
xy((empty(x) empty(y) x=y)
• Note: this formula quantifies over sets!
9/27/2010 Michael P. Frank / Kees van Deemter 13
Module #3 - Sets
There‟s only one empty set
Prove that xy((empty(x) empty(y)) x=y)
Proof by Reductio ad Absurdum:
• Suppose there existed a and b
such that empty(a) and empty(b).
• Thus, x(xa) x(xb)
• Suppose ab. This would mean that either
x(xa xb) or x(xb xa)
• But the first case cannot hold, for x(xa). The
second case cannot hold, for x(xb)
• Contradiction, so QED
9/27/2010 Michael P. Frank / Kees van Deemter 14
Module #3 - Sets
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.
• = {} = {x|xx} = ... = {x|False}
• Any set containing exactly one element is
called a singleton
9/27/2010 Michael P. Frank / Kees van Deemter 15
Module #3 - Sets
Subset and Superset Relations
• ST (“S is a subset of T”) means that every
element of S is also an element of T.
• ST :def x (xS xT)
• What do you think about these?
– S ?
– SS ?
9/27/2010 Michael P. Frank / Kees van Deemter 16
Module #3 - Sets
Subset and Superset Relations
• ST (“S is a subset of T”) means that every
element of S is also an element of T.
• ST :def x (xS xT)
• What do you think about these?
– S ? Yes
– SS ? Yes
9/27/2010 Michael P. Frank / Kees van Deemter 17
Module #3 - Sets
Subset and Superset Relations
• More notation:
• ST (“S is a superset of T”) :def TS.
Note S=T ST ST.
• S T :def (ST), i.e. x(xS xT)
/
9/27/2010 Michael P. Frank / Kees van Deemter 18
Module #3 - Sets
Proper (Strict) Subsets & Supersets
• ST (“S is a proper subset of T”) means that
ST but T. S
/
Example:{1,2} {1,2,3}
We have {1,2,3} {1,2,3},
but not {1,2,3} {1,2,3}
9/27/2010 Michael P. Frank / Kees van Deemter 19
Module #3 - Sets
Sets Are Objects, Too!
• The elements of a set may themselves be
sets.
• E.g. let S={x | x {1,2,3}}
then S = …
9/27/2010 Michael P. Frank / Kees van Deemter 20
Module #3 - Sets
Sets Are Objects, Too!
• The objects that are elements of a set may
themselves be sets.
• E.g. let S={x | x {1,2,3}}
then S={,
{1}, {2}, {3},
{1,2}, {1,3}, {2,3},
{1,2,3}}
• Note that 1 {1} {{1}}
9/27/2010 Michael P. Frank / Kees van Deemter 21
Module #3 - Sets
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.
9/27/2010 Michael P. Frank / Kees van Deemter 22
Module #3 - Sets
The Power Set Operation
• The power set P(S) of a set S is the set of all
subsets of S. P(S) :≡ {x | xS}.
• E.g. P({a,b}) = {, {a}, {b}, {a,b}}.
• Sometimes P(S) is written 2S, because
|P(S)| = 2|S|.
• It turns out S:|P(S)|>|S|, e.g. |P(N)| > |N|.
There are different sizes of infinite sets!
9/27/2010 Michael P. Frank / Kees van Deemter 23
Module #3 - Sets
Review: Set Notations So Far
• Set enumeration {a, b, c}
and set-builder {x|P(x)}.
• relation, and the empty set .
• Set relations =, , , , , , etc.
• Venn diagrams.
• Cardinality |S| and infinite sets N, Z, R.
• Power sets P(S).
9/27/2010 Michael P. Frank / Kees van Deemter 24
Module #3 - Sets
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!
9/27/2010 Michael P. Frank / Kees van Deemter 25
Module #3 - Sets
This version of
Set Theory is inconsistent
Russell’s paradox:
• Consider the set that corresponds with the
predicate x x :
S = {x | xx }.
• Now ask: is SS?
9/27/2010 Michael P. Frank / Kees van Deemter 26
Module #3 - Sets
Russell‟s paradox
• Let S = {x | xx }. Is SS?
• If SS, then S is one of those objects x for which
xx. In other words, SS
By Reductio, we have SS
• If SS, then S is not one of those objects x for
which xx. In other words, SS
By Reductio, we have SS
• We conclude that both SS nor SS
• Paradox! (There‟s no assumption that we can
blame, so we cannot Reductio again)
9/27/2010 Michael P. Frank / Kees van Deemter 27
Module #3 - Sets
• To avoid inconsistency, set theory must
somehow change
Bertrand Russell
1872-1970
9/27/2010 Michael P. Frank / Kees van Deemter 28
Module #3 - Sets
( 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. )
9/27/2010 Michael P. Frank / Kees van Deemter 29
Module #3 - Sets
Ordered n-tuples
• These are like sets, except that duplicates
matter, and the order makes a difference.
• For nN, an ordered n-tuple or a sequence
of length n is written (a1, a2, …, an). Its first
element is a1, etc.
Contrast with
• Note that (1, 2) (2, 1) (2, 1, 1). sets: {...}
• Empty sequence, singlets, pairs, triples, …,
n-tuples.
9/27/2010 Michael P. Frank / Kees van Deemter 30
Module #3 - Sets
• n-tuples have many applications. For
example,
9/27/2010 Michael P. Frank / Kees van Deemter 31
Module #3 - Sets
• Relations are often spelled out by means of n-
tuples. E.g., here are two 2-place relations:
< = { (0,1), (1,2), (0,2), …) }
Like-to-watch =
{(John,news),(Mary,soap),(Ellen,movies)}
• 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
9/27/2010 Michael P. Frank / Kees van Deemter 32
Module #3 - Sets
Cartesian Products of Sets
• For sets A, B, their Cartesian product
AB : {(a, b) | aA bB }.
• E.g. {a,b}{1,2} = {(a,1),(a,2),(b,1),(b,2)}
• {John,Mary,Ellen}x{News,Soap}=
René Descartes
9/27/2010 Michael P. Frank / Kees van Deemter (1596-1650)33
Module #3 - Sets
Cartesian Products of Sets
• For sets A, B, their Cartesian product
AB : {(a, b) | aA bB }.
• E.g. {a,b}{1,2} = {(a,1),(a,2),(b,1),(b,2)}
• {John,Mary,Ellen}x{News,Soap}=
{(John,News),(Mary,News),(Ellen,News),
(John,Soap),(Mary,Soap),(Ellen,Soap)}
• If R is a relation between A and B then
RAxB
9/27/2010 Michael P. Frank / Kees van Deemter 34
Module #3 - Sets
Cartesian Products of Sets
• Note that
– for finite A, B, |AB| = |A|.|B|
– the Cartesian product is not commutative: i.e.,
AB: AB=BA.
– notation extends naturally to A1 A2 … An
9/27/2010 Michael P. Frank / Kees van Deemter 35
Module #3 - Sets
Review of §1.6
• Sets S, T, U… Special sets N, Z, R.
• Set notations {a,b,...}, {x|P(x)}…
• Set relation operators xS, ST, ST, S=T,
ST, ST. (These form propositions.)
• Finite vs. infinite sets.
• Set operations |S|, P(S), ST.
• Next up: §1.5: More set ops: , , .
9/27/2010 Michael P. Frank / Kees van Deemter 36
Module #3 - Sets
Start §1.7: The Union Operator
• For sets A, B, theirnion AB is the set
containing all elements that are either in A,
or (“”) in B (or, of course, in both).
• Formally, A,B: AB = {x | xA xB}.
• Note that AB is a superset of both A and
B (in fact, it is the smallest such superset):
A, B: (AB A) (AB B)
9/27/2010 Michael P. Frank / Kees van Deemter 37
Module #3 - Sets
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}
9/27/2010 Michael P. Frank / Kees van Deemter 38
Module #3 - Sets
The Intersection Operator
• For sets A, B, their intersection AB is the
set containing all elements that are
simultaneously in A and (“”) in B.
• Formally, A,B: AB={x | xA xB}.
• Note that AB is a subset of both A and B
(in fact it is the largest such subset):
A, B: (AB A) (AB B)
9/27/2010 Michael P. Frank / Kees van Deemter 39
Module #3 - Sets
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.”
9/27/2010 Michael P. Frank / Kees van Deemter 40
Module #3 - Sets
Disjointness
• Two sets A, B are called
disjoint (i.e., not joined)
iff their intersection is
empty. (AB=)
• Example: the set of even
integers is disjoint with
the set of odd integers.
9/27/2010 Michael P. Frank / Kees van Deemter 41
Module #3 - Sets
Inclusion-Exclusion Principle
• How many elements are in AB?
Can you think of a general formula?
(Express in terms of |A| and |B| and
whatever else you need.)
9/27/2010 Michael P. Frank / Kees van Deemter 42
Module #3 - Sets
Inclusion-Exclusion Principle
• How many elements are in AB?
|AB| = |A| |B| |AB|
• Example: How many students are on our
class email list? Consider set E I M,
I = {s | s turned in an information sheet}
M = {s | s sent the TAs their email address}
• Some students may have done both!
|E| = |IM| = |I| |M| |IM|
9/27/2010 Michael P. Frank / Kees van Deemter 43
Module #3 - Sets
Set Difference
• For sets A, B, the difference of A and B,
written AB, is the set of all elements that
are in A but not B. Formally:
A B : x xA xB
• Also called:
The complement of B with respect to A.
9/27/2010 Michael P. Frank / Kees van Deemter 44
Module #3 - Sets
Set Difference Examples
• {1,2,3,4,5,6} {2,3,5,7,9,11} =
___________
{1,4,6}
• 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}
9/27/2010 Michael P. Frank / Kees van Deemter 45
Module #3 - Sets
Set Difference - Venn Diagram
• A−B is what‟s left after B
“takes a bite out of A”
Chomp!
Set
AB
Set A Set B
9/27/2010 Michael P. Frank / Kees van Deemter 46
Module #3 - Sets
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 AU, the complement of A,
written A, is the complement of A w.r.t. U,
i.e., it is UA.
• E.g., If U=N, {3,5} {0,1,2,4,6,7,...}
9/27/2010 Michael P. Frank / Kees van Deemter 47
Module #3 - Sets
Set Identities
• A
9/27/2010 Michael P. Frank / Kees van Deemter 48
Module #3 - Sets
Set Identities
• A = A
• AU =
9/27/2010 Michael P. Frank / Kees van Deemter 49
Module #3 - Sets
Set Identities
• A = A
• AU = A
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Module #3 - Sets
Set Identities
• A = A = AU
• AU = U
A =
(A )
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Module #3 - Sets
Set Identities
• A = A = AU
• AU = U
A =
• AA = A = AA
• AB = BA ( A) A
AB = BA
• A(BC)=(AB)C A(BC)=(AB)C
9/27/2010 Michael P. Frank / Kees van Deemter 52
Module #3 - Sets
Have you seen
similar patterns before?
9/27/2010 Michael P. Frank / Kees van Deemter 53
Module #3 - Sets
Read: := , := , :=F, U:=T
• A = A = AU
• AU = U , A =
• AA = A = AA
• AB = BA , AB = BA
• A(BC)=(AB)C ,
A(BC)=(AB)C
9/27/2010 Michael P. Frank / Kees van Deemter 54
Module #3 - Sets
Set Identities
(don‟t worry about their names)
• Identity: A = A = AU
• Domination: AU = U , A =
• Idempotent: AA = A = AA
• Double complement: ( A) A
• Commutative: AB = BA , AB = BA
• Associative: A(BC)=(AB)C ,
A(BC)=(AB)C
9/27/2010 Michael P. Frank / Kees van Deemter 55
Module #3 - Sets
DeMorgan‟s Law for Sets
• Exactly analogous to (and provable from)
DeMorgan‟s Law for propositions.
A B A B
A B A B
9/27/2010 Michael P. Frank / Kees van Deemter 56
Module #3 - Sets
( 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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Module #3 - Sets
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. Use equivalence laws
2. Prove E1 E2 and E2 E1 separately.
3. Use a membership table.
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Module #3 - Sets
Method 2: Mutual subsets
Example: Show A(BC)=(AB)(AC).
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Module #3 - Sets
Method 2: Mutual subsets
Example: Show A(BC)=(AB)(AC).
• Part 1: Show A(BC)(AB)(AC).
– Assume xA(BC), & show x(AB)(AC).
– We know that xA, and either xB or xC.
• Case 1: xB. Then xAB, so x(AB)(AC).
• Case 2: xC. Then xAC , so x(AB)(AC).
– Therefore, x(AB)(AC).
– Therefore, A(BC)(AB)(AC).
• Part 2: Show (AB)(AC) A(BC).
(analogous)
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Module #3 - Sets
Method 2: 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(BC)(AB)(AC).
Suppose xA (xB xC).
Prove (xA xB) (xA xC).
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Module #3 - Sets
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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Module #3 - Sets
Membership Table Example
Prove (AB)B = AB.
A B AB (AB)B AB
0 0 0 0 0
0 1 1 0 0
1 0 1 1 1
1 1 1 0 0
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Module #3 - Sets
Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB ( A B ) C A C B C (AC)(BC)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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Module #3 - Sets
Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB (AB)C AC BC (AC)(BC)
000
001
010 1
011
100 1
101
110 1
111
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Module #3 - Sets
Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB (AB)C AC BC (AC)(BC)
000
001
010 1 1
011
100 1 1
101
110 1 1 1
111
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Module #3 - Sets
Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB (AB)C AC BC (AC)(BC)
000
001
010 1 1 1
011
100 1 1 1
101
110 1 1 1 1
111
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Module #3 - Sets
Review of §1.6-1.7
• Sets S, T, U… Special sets N, Z, R.
• Set notations {a,b,...}, {x|P(x)}…
• Relations xS, ST, ST, S=T, ST, ST.
• Operations |S|, P(S), , , , , S
• Set equality proof techniques
9/27/2010 Michael P. Frank / Kees van Deemter 68
Module #3 - Sets
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,
X={A | P(A)} (for some property P).
(This is just like using when adding up
large or variable numbers of numbers)
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Module #3 - Sets
Generalized Union
• Binary union operator: AB
• n-ary union:
A1A2…An : ((…((A1 A2) …) An)
(grouping & order is irrelevant)
n
• “Big U” notation: A i 1
i
• Or for infinite sets of sets: A
A X
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Module #3 - Sets
Generalized Intersection
• Binary intersection operator: AB
• n-ary intersection:
A1A2…An((…((A1A2)…)An)
(grouping & order is irrelevant)
n
• “Big Arch” notation: A i 1
i
• Or for infinite sets of sets: A
A X
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Module #3 - Sets
(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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Module #3 - Sets
Representations
– Sets: 0:, 1:{0}, 2:{0,1}, 3:{0,1,2}, …
– General: n : {x N : x<n}
– Can you write 3 more fully?
0=
1 = {}
2 = {,{}}
3 = {,{},{,{}}}
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Module #3 - Sets
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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Module #3 - Sets
Representing Sets with Bit Strings
For an enumerable u.d. U with ordering
x1, x2, …, represent a finite set SU as the
finite bit string B=b1b2…bn where
i: xiS (i<n bi=1).
E.g. U=N, S={2,3,5,7,11}, B=001101010001.
In this representation, the set operators
“”, “”, “-” are implemented directly by
bitwise OR, AND, NOT!
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Module #3 - Sets
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
9/27/2010 Michael P. Frank / Kees van Deemter 76
Module #3 - Sets
• We now know enough about sets to move
on to relations between sets, and functions
from one set to another
9/27/2010 Michael P. Frank / Kees van Deemter 77
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