Introduction to set theory with application

A set is a structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
Set theory deals with operations between, relations among, and statements about sets

1. 1
Set Theory
Rosen 6
th
ed., §2.1-2.2

2. 2
Introduction to Set Theory
•A setis a structure, representing an
unorderedcollection (group, plurality) of
zero or more distinct(different) objects.
•Set theory deals with operations between,
relations among, and statements about sets.

3. 3
Basic notations for sets
•For sets, we’ll use variables S, T, U, …
•We can denote a set Sin 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.
•Setbuilder notation: For any proposition P(x) over
any universe of discourse, {x|P(x)} is the set of all
x such that P(x).
e.g., {x| xis an integer where x>0 and x<5 }

4. 4
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} =
{b, c, a} = {c, a, b} = {c, b, a}.
•All elements are distinct(unequal);
multiple listings make no difference!
–{a, b, c} = {a, a, b, a, b, c, c, c, c}.
–This set contains at most 3 elements!

5. 5
Definition of Set Equality
•Two sets are declared to be equal if and only if
they contain exactly the sameelements.
•In particular, it does not matter how the set is
defined or denoted.
•For example: The set {1, 2, 3, 4} =
{x| xis an integer where x>0 and x<5 } =
{x| xis a positive integer whose square
is >0 and <25}

6. 6
Infinite Sets
•Conceptually, 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…
•Infinite sets come in different sizes!

7. 7
Venn Diagrams

8. 8
Basic Set Relations: Member of
•xS (“xis in S”)is the proposition that object xis
an lementor memberof set S.
–e.g.3N, “a”{x | xis a letter of the alphabet}
•Can define set equalityin terms of relation:
S,T: S=T (x: xSxT)
“Two sets are equal iffthey have all the same
members.”
•xS :(xS) “xis not in S”

9. 9
The Empty Set
•(“null”, “the empty set”) is the unique set
that contains no elements whatsoever.
•= {} = {x|False}
•No matter the domain of discourse,
we have the axiom
x: x.

10. 10
Subset and Superset Relations
•ST(“Sis a subset of T”) means that every
element of Sis also an element of T.
•ST x (xSxT)
•S, SS.
•ST(“Sis a superset of T”) means TS.
•Note S=TSTST.
• means (ST), i.e.x(xSxT)TS/

11. 11
Proper (Strict) Subsets & Supersets
•ST (“Sis a proper subset of T”) means that
ST but . Similar for ST.ST/
S
T
Venn Diagram equivalent of ST
Example:
{1,2} 
{1,2,3}

12. 12
Sets Are Objects, Too!
•The objects that are elements of a set may
themselvesbe 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}} !!!!

13. 13
Cardinality and Finiteness
•|S| (read “the cardinalityof S”) is a measure
of how many different elements Shas.
•E.g., ||=0, |{1,2,3}| = 3, |{a,b}| = 2,
|{{1,2,3},{4,5}}| = ____
•We say Sis infiniteif it is not finite.
•What are some infinite sets we’ve seen?

14. 14
The Power SetOperation
•The power setP(S) of a set Sis 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 2
S
.
Note that for finite S, |P(S)| = 2
|S|
.
•It turns out that |P(N)| > |N|.
There are different sizes of infinite sets!

15. 15
Ordered n-tuples
•For nN, an ordered n-tupleor a sequence
oflength nis written (a
1, a
2, …, a
n). The
firstelement is a
1, etc.
•These are like sets, except that duplicates
matter, and the order makes a difference.
•Note (1, 2) (2, 1) (2, 1, 1).
•Empty sequence, singlets, pairs, triples,
quadruples, quintuples, …, n-tuples.

16. 16
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)}
•Note that for finite A, B, |AB|=|A||B|.
•Note that the Cartesian product is not
commutative: AB: AB =BA.
•Extends to A
1A
2… A
n...

17. 17
The Union Operator
•For sets A, B, theirunionABis 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 contains all the elements of
Aandit contains all the elements of B:
A, B: (AB A) (AB B)

18. 18
•{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}
Union Examples

19. 19
The Intersection Operator
•For sets A, B, their intersectionABis 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 Aandit is a
subset of B:
A, B: (AB A) (AB B)

20. 20
•{a,b,c}{2,3} = ___
•{2,4,6}{3,4,5}= ______
Intersection Examples

{4}

21. 21
Disjointedness
•Two sets A, Bare called
disjoint(i.e., unjoined)
iff their intersection is
empty. (AB=)
•Example: the set of even
integers is disjoint with
the set of odd integers.
Help, I’ve
been
disjointed!

22. 22
Inclusion-Exclusion Principle
•How many elements are in AB?
|AB|= |A| |B| |AB|
•Example:
{2,3,5}{3,5,7}= {2,3,5,3,5,7} ={2,3,5,7}

23. 23
Set Difference
•For sets A, B, the differenceof A and B,
written AB, is the set of all elements that
are in Abut not B.
•A B :x xA xB
xxAxB
•Also called:
The complementofBwith respect toA.

24. 24
Set Difference Examples
•{1,2,3,4,5,6} {2,3,5,7,9,11} =
___________
•Z N {… , -1, 0, 1, 2, … } {0, 1, … }
= {x | xis an integer but not a nat. #}
= {x|xis a negative integer}
= {… , -3, -2, -1}
{1,4,6}

25. 25
Set Difference -Venn Diagram
•A-Bis what’s left after B
“takes a bite out of A”
Set A Set B
Set
AB
Chomp!

26. 26
Set Complements
•The universe of discoursecan itself be
considered a set, call it U.
•The complementof A, written , is the
complement of Aw.r.t. U, i.e.,it is UA.
•E.g., If U=N, A ,...}7,6,4,2,1,0{}5,3{

27. 27
More on Set Complements
•An equivalent definition, when Uis clear:}|{ AxxA 
A
UA

28. 28
Set Identities
•Identity: A=AAU=A
•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)CAA)(

29. 29
DeMorgan’s Law for Sets
•Exactly analogous to (and derivable from)
DeMorgan’s Law for propositions.BABA
BABA



30. 30
Proving Set Identities
To prove statements about sets, of the form
E
1= E
2(where Es are set expressions), here
are three useful techniques:
•Prove E
1E
2andE
2E
1separately.
•Use logical equivalences.
•Use a membership table.

31. 31
Method 1: Mutual subsets
Example: Show A(BC)=(AB)(AC).
•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Bor 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).
•Show (AB)(AC) A(BC). …

32. 32
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.

33. 33
Membership Table Example
Prove (AB)B = AB.AABBAABB((AABB))BBAABB
000 0 0
011 0 0
101 1 1
111 0 0

34. 34
Membership Table Exercise
Prove (AB)C= (AC)(BC).ABCAABB((AABB))CCAACCBBCC((AACC))((BBCC))
000
001
010
011
100
101
110
111

35. 35
Generalized Union
•Binary union operator: AB
•n-ary union:
AA
2…A
n:((…((A
1A
2)…)A
n)
(grouping & order is irrelevant)
•“Big U” notation:
•Or for infinite sets of sets:
n
i
iA
1 
XA
A


36. 36
Generalized Intersection
•Binary intersection operator: AB
•n-ary intersection:
AA
2…A
n((…((A
1A
2)…)A
n)
(grouping & order is irrelevant)
•“Big Arch” notation:
•Or for infinite sets of sets:
n
i
iA
1 
XA
A
