Your Federal Quarterly Tax Payments are due April 15th

# University of Aberdeen, Computing Science CS3511 Discrete by wgv13363

VIEWS: 4 PAGES: 77

• pg 1
```									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
• 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
• 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 :
S=T :def x: xS  xT
“Two sets are equal iff they have
the same members.”
• Notation: xS :def (xS)

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(xS).
• 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)
• 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)

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|xx} = ... = {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
• 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 ?

9/27/2010             Michael P. Frank / Kees van Deemter   16
Module #3 - Sets

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

9/27/2010                 Michael P. Frank / Kees van Deemter   17
Module #3 - Sets

Subset and Superset Relations
• More notation:
• ST (“S is a superset of T”) :def TS.
Note S=T  ST ST.
• S  T :def (ST), i.e. x(xS  xT)
/

9/27/2010            Michael P. Frank / Kees van Deemter   18
Module #3 - Sets

Proper (Strict) Subsets & Supersets

• ST (“S is a proper subset of T”) means that
ST 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 | xS}.
• 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

• Consider the set that corresponds with the
predicate x  x :
S = {x | xx }.

9/27/2010                Michael P. Frank / Kees van Deemter   26
Module #3 - Sets

• 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S
By 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S
By Reductio, we have SS
• We conclude that both SS nor SS
• 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 nN, 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
AB : {(a, b) | aA  bB }.
• 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
AB : {(a, b) | aA  bB }.
• 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
RAxB

9/27/2010             Michael P. Frank / Kees van Deemter   34
Module #3 - Sets

Cartesian Products of Sets
• Note that
– for finite A, B, |AB| = |A|.|B|
– the Cartesian product is not commutative: i.e.,
AB: AB=BA.
– 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 xS, ST, ST, S=T,
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: , , .

9/27/2010            Michael P. Frank / Kees van Deemter   36
Module #3 - Sets

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)

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 AB is the
set containing all elements that are
simultaneously in A and (“”) in B.
• Formally, A,B: AB={x | 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)

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
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. (AB=)
• 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 AB?
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 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 sheet}
M = {s | s sent the TAs their email address}
• Some students may have done both!
|E| = |IM| = |I|  |M|  |IM|
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 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.

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
AB

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 AU, the complement of A,
written A, is the complement of A w.r.t. U,
i.e., it is UA.
• 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
• AU =

9/27/2010          Michael P. Frank / Kees van Deemter   49
Module #3 - Sets

Set Identities
• A = A
• AU = A

9/27/2010          Michael P. Frank / Kees van Deemter   50
Module #3 - Sets

Set Identities
• A = A = AU
• AU = U
A = 
(A ) 

9/27/2010          Michael P. Frank / Kees van Deemter   51
Module #3 - Sets

Set Identities
• A = A = AU
• AU = U
A = 
• AA = A = AA
• AB = BA       ( A)  A
AB = BA
• A(BC)=(AB)C A(BC)=(AB)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 = 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

9/27/2010            Michael P. Frank / Kees van Deemter   54
Module #3 - Sets

Set Identities
•    Identity:    A = A = AU
•    Domination: AU = U , A = 
•    Idempotent: AA = A = AA
•    Double complement: ( A)  A
•    Commutative: AB = BA , AB = BA
•    Associative: A(BC)=(AB)C ,
A(BC)=(AB)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.                                )

9/27/2010                   Michael P. Frank / Kees van Deemter       57
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.

9/27/2010              Michael P. Frank / Kees van Deemter   58
Module #3 - Sets

Method 2: Mutual subsets
Example: Show A(BC)=(AB)(AC).

9/27/2010          Michael P. Frank / Kees van Deemter   59
Module #3 - Sets

Method 2: 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)

9/27/2010                 Michael P. Frank / Kees van Deemter   60
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(BC)(AB)(AC).
Suppose xA  (xB  xC).
Prove (xA  xB)  (xA  xC).

9/27/2010             Michael P. Frank / Kees van Deemter   61
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.

9/27/2010              Michael P. Frank / Kees van Deemter   62
Module #3 - Sets

Membership Table Example
Prove (AB)B = AB.
A      B AB (AB)B AB
0      0  0     0     0
0      1  1     0     0
1      0  1     1     1
1      1  1     0     0

9/27/2010             Michael P. Frank / Kees van Deemter   63
Module #3 - Sets

Membership Table Exercise
Prove (AB)C = (AC)(BC).
A   B   C AB ( A B )  C A C            B C          (AC)(BC)
0   0   0
0   0   1
0   1   0
0   1   1
1   0   0
1   0   1
1   1   0
1   1   1

9/27/2010                      Michael P. Frank / Kees van Deemter                 64
Module #3 - Sets

Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB (AB)C AC              BC            (AC)(BC)
000
001
010          1
011
100          1
101
110          1
111

9/27/2010               Michael P. Frank / Kees van Deemter                 65
Module #3 - Sets

Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB (AB)C AC              BC            (AC)(BC)
000
001
010          1                        1
011
100          1     1
101
110          1     1                  1
111

9/27/2010               Michael P. Frank / Kees van Deemter                 66
Module #3 - Sets

Membership Table Exercise
Prove (AB)C = (AC)(BC).
A B C AB (AB)C AC               BC           (AC)(BC)
000
001
010          1                         1              1
011
100          1     1                                  1
101
110          1     1                   1              1
111

9/27/2010               Michael P. Frank / Kees van Deemter                 67
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 xS, ST, ST, S=T, ST, ST.
•    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)

9/27/2010                Michael P. Frank / Kees van Deemter   69
Module #3 - Sets

Generalized Union
• Binary union operator: AB
• n-ary union:
A1A2…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

9/27/2010              Michael P. Frank / Kees van Deemter   70
Module #3 - Sets

Generalized Intersection
• Binary intersection operator: AB
• n-ary intersection:
A1A2…An((…((A1A2)…)An)
(grouping & order is irrelevant)
n
• “Big Arch” notation:     A      i 1
i

• Or for infinite sets of sets:                 A
A X

9/27/2010              Michael P. Frank / Kees van Deemter    71
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?

9/27/2010                Michael P. Frank / Kees van Deemter   72
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 = {,{},{,{}}}

9/27/2010                Michael P. Frank / Kees van Deemter   73
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.                                          )
9/27/2010                Michael P. Frank / Kees van Deemter       74
Module #3 - Sets

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 B=b1b2…bn where
i: xiS  (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!
9/27/2010             Michael P. Frank / Kees van Deemter   75
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

```
To top