1. COT 3100, Spring 2001 Applications of Discrete Structures
Section #1089X - MWF 4th period
Dr. Michael P. Frank Lecture #8 Fri., Jan. 26, 2001
1/26/01
Lecture #8

2. Administrivia Today: Returning graded HW#1 papers.
Quiz now on HW#2 material (§§ ).
Lecture: finish §1.4, do §1.5, Set Operations
in E121: Distinguished lecturer Christos Papadimitriou on “Algorithmic Problems Related to the Internet.”
1/26/01
Lecture #8

3. Review: Set Notations So Far
Variable objects x, y, z; sets S, T, U.
Literal set {a, b, c} and set-builder {x|P(x)}.
 relational operator, and the empty set .
Set relations =, , , , , , etc.
Venn diagrams.
Cardinality |S| and infinite sets N, Z, R.
Power sets P(S).
1/26/01
Lecture #8

4. Naïve Set Theory is Inconsistent
Some naïve descriptions of sets lead to structures that are not well-defined. (That do not have consistent properties.)
These “sets” mathematically cannot exist.
E.g. let S = {x | xx }. Is SS?
Consistent set theories must restrict the language that can be used to describe sets.
For purposes of this class, don’t worry about it!
1/26/01
Lecture #8

5. Ordered n-tuples
Like sets, but 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). The first element is a1, etc.
Note (1, 2)  (2, 1)  (2, 1, 1).
Empty sequence, singlets, pairs, triples, quadruples, quintuples, …, n-tuples.
1/26/01
Lecture #8

6. 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 A1  A2  …  An...
1/26/01
Lecture #8

7. End of §1.4 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: , , .
1/26/01
Lecture #8

8. Start §1.5: 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 contains all the elements of A and it contains all the elements of B: A, B: (ABA)  (ABB)
1/26/01
Lecture #8

9. Union Examples Required Form 2 5 3 7 {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}
Required Form
Think “The United States of America includes every person who worked in any state last year.” (How the IRS sees it...) 2 3 5 7
1/26/01
Lecture #8

10. 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 A and it is a subset of B: A, B: (ABA)  (ABB)
1/26/01
Lecture #8

11. Intersection Examples
{a,b,c}{2,3} = ___
{2,4,6}{3,4,5} = ______ 
{4} 2 3 5 6 4
Think “The intersection of University and 13th St. is that portion of road surface that lies on both streets.”
1/26/01
Lecture #8

12. Help, I’ve been disjointed!
Disjointedness
Help, I’ve been disjointed!
Two sets A, B are 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.
1/26/01
Lecture #8

13. Inclusion-Exclusion Principle
How many elements are in AB? |AB| = |A|  |B|  |AB|
Example: How many students are on our class list? Consider set E  I  M, I = {s | s turned in an information sheet} M = {s | s signed up through majordomo}
Some students did both! |E| = |IM| = |I|  |M|  |IM|
Subtract out items
in intersection, to
compensate for
double-counting them!
1/26/01
Lecture #8

14. 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.
A  B  x  xA  xB  x   xA  xB  
Also called: The complement of B with respect to A.
1/26/01
Lecture #8

15. 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}
1/26/01
Lecture #8

16. 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
1/26/01
Lecture #8

17. 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 U=N,
1/26/01
Lecture #8

18. More on Set Complements
An equivalent definition, when U is clear: A U
1/26/01
Lecture #8

19. Set Identities Identity: A=A A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)C
1/26/01
Lecture #8

20. DeMorgan’s Law for Sets
Exactly analogous to (and derivable from) DeMorgan’s Law for propositions.
1/26/01
Lecture #8

21. No More Slides
It is now 2:15 am and Dr. Frank is getting too sleepy to continue making slides tonight...
1/26/01
Lecture #8