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CIS160Mathematical Foundations of

Computer ScienceSome Notes

Jean GallierDepartment of Computer and Information Science

University of PennsylvaniaPhiladelphia, PA 19104, USAe-mail: jean@cis.upenn.edu

c� Jean GallierPlease, do not reproduce without permission of the author

December 15, 2009

2

Contents

1 Mathematical Reasoning, Proof Princi-ples and Logic 71.1 Problems, Motivations . . . . . . . . . . . 71.2 Inference Rules, Deductions, The Proof

Systems N⇒m and NG⇒m . . . . . . . . . . 19

1.3 Adding ∧, ∨, ⊥; The Proof SystemsN⇒,∧,∨,⊥

c and NG⇒,∧,∨,⊥c . . . . . . . . . 46

1.4 Clearing Up Differences Between Rules In-volving ⊥ . . . . . . . . . . . . . . . . . . 71

1.5 De Morgan Laws and Other Rules of Clas-sical Logic . . . . . . . . . . . . . . . . . 80

1.6 Formal Versus Informal Proofs; Some Ex-amples . . . . . . . . . . . . . . . . . . . 85

1.7 Truth Values Semantics for Classical Logic 1031.8 Adding Quantifiers; The Proof Systems

N⇒,∧,∨,∀,∃,⊥c , NG⇒,∧,∨,∀,∃,⊥

c . . . . . . . . 1131.9 First-Order Theories . . . . . . . . . . . . 1481.10 Basics Concepts of Set Theory . . . . . . 166

3

4 CONTENTS

2 Relations, Functions, Partial Functions 1932.1 What is a Function? . . . . . . . . . . . . 1932.2 Ordered Pairs, Cartesian Products, Rela-

tions, etc. . . . . . . . . . . . . . . . . . . 2032.3 Induction Principles on N . . . . . . . . . 2192.4 Composition of Relations and Functions . 2412.5 Recursion on N . . . . . . . . . . . . . . . 2452.6 Inverses of Functions and Relations . . . . 2502.7 Injections, Surjections, Bijections, Permu-

tations . . . . . . . . . . . . . . . . . . . 2582.8 Direct Image and Inverse Image . . . . . . 2682.9 Equinumerosity; Pigeonhole Principle;

Schroder–Bernstein . . . . . . . . . . . . . 2742.10 An Amazing Surjection: Hilbert’s Space

Filling Curve . . . . . . . . . . . . . . . . 2982.11 Strings, Multisets, Indexed Families . . . . 302

3 Graphs, Part I: Basic Notions 3193.1 Why Graphs? Some Motivations . . . . . 3193.2 Directed Graphs . . . . . . . . . . . . . . 3243.3 Path in Digraphs; Strongly Connected Com-

ponents . . . . . . . . . . . . . . . . . . . 3363.4 Undirected Graphs, Chains, Cycles, Con-

nectivity . . . . . . . . . . . . . . . . . . 3623.5 Trees and Arborescences . . . . . . . . . . 380

CONTENTS 5

3.6 Minimum (or Maximum) Weight SpanningTrees . . . . . . . . . . . . . . . . . . . . 389

4 Some Counting Problems; MultinomialCoefficients 4014.1 Counting Permutations and Functions . . 4014.2 Counting Subsets of Size k; Multinomial

Coefficients . . . . . . . . . . . . . . . . . 4074.3 Some Properties of the Binomial Coefficients4234.4 The Inclusion-Exclusion Principle . . . . . 446

5 Partial Orders, Equivalence Relations, Lat-tices 4595.1 Partial Orders . . . . . . . . . . . . . . . 4595.2 Lattices and Tarski’s Fixed Point Theorem 4785.3 Well-Founded Orderings and Complete In-

duction . . . . . . . . . . . . . . . . . . . 4875.4 Unique Prime Factorization in Z and GCD’s5025.5 Equivalence Relations and Partitions . . . 5215.6 Transitive Closure, Reflexive and Transi-

tive Closure . . . . . . . . . . . . . . . . 5335.7 Fibonacci and Lucas Numbers; Mersenne

Primes . . . . . . . . . . . . . . . . . . . 5365.8 Distributive Lattices, Boolean Algebras, Heyt-

ing Algebras . . . . . . . . . . . . . . . . 570

6 CONTENTS

Chapter 1

Mathematical Reasoning, ProofPrinciples and Logic

1.1 Problems, Motivations

Problem 1. Find formulae for the sums

1 + 2 + 3 + · · · + n = ?

12 + 22 + 32 + · · · + n2 = ?

13 + 23 + 33 + · · · + n3 = ?

· · · · · · · · · · · ·1k + 2k + 3k + · · · + nk = ?

Jacob Bernoulli (1654-1705) discovered the formulae listedbelow:

7

8 CHAPTER 1. MATHEMATICAL REASONING, PROOF PRINCIPLES AND LOGIC

IfSk(n) = 1k + 2k + 3k + · · · + nk

then

S0(n) = 1n

S1(n) =1

2n2 +

1

2n

S2(n) =1

3n3 +

1

2n2 +

1

6n

S3(n) =1

4n4 +

1

2n3 +

1

4n2

S4(n) =1

5n5 +

1

2n4 +

1

3n3 − 1

30n

S5(n) =1

6n6 +

1

2n5 +

5

12n4 − 1

12n2

S6(n) =1

7n7 +

1

2n6 +

1

2n5 − 1

6n3 +

1

42n

S7(n) =1

8n8 +

1

2n7 +

7

12n6 − 7

24n4 +

1

12n2

S8(n) =1

9n9 +

1

2n8 +

2

3n7 − 7

15n5 +

2

9n3 − 1

30n

S9(n) =1

10n10 +

1

2n9 +

3

4n8 − 7

10n6 +

1

2n4 − 3

20n2

S10(n) =1

11n11 +

1

2n10 +

5

6n9 − n7 + n5 − 1

2n3 +

5

66n

Is there a pattern?

What are the mysterious numbers

11

2

1

60 − 1

300

1

420 − 1

300

5

66?

1.1. PROBLEMS, MOTIVATIONS 9

The next two are

0 − 691

2730Why?

It turns out that the answer has to do with the Bernoullipolynomials , Bk(x), with

Bk(x) =k�

i=0

�k

i

�xk−iBi,

where the Bi are the Bernoulli numbers .

There are various ways of computing the Bernoulli num-bers, including some recurrence formulae.

Amazingly, the Bernoulli numbers show up in very differ-ent areas of mathematics, in particular, algebraic topol-ogy!

10 CHAPTER 1. MATHEMATICAL REASONING, PROOF PRINCIPLES AND LOGIC

Figure 1.1: Funny spheres (in 3D)

Figure 1.2: A plane (granite slab!)

Problem 2. Prove that a sphere and a plane in 3D havethe same number of points .

More precisely, find a one-to-one and onto mapping of thesphere onto the plane (a bijection)

Actually, there are also bijections between the sphere anda (finite) rectangle, with or without its boundary!

1.1. PROBLEMS, MOTIVATIONS 11

Problem 3. Counting the number of derangements ofn elements.

A permutation of the set {1, 2, . . . , n} is any one-to-onefunction, f , of {1, 2, . . . , n} into itself. A permutation ischaracterized by its image: {f (1), f(2), . . . , f (n)}.

For example, {3, 1, 4, 2} is a permutation of {1, 2, 3, 4}.

It is easy to show that there are n! = n · (n− 1) · · · 3 · 2distinct permutations of n elements.

A derangements is a permutation that leaves no elementfixed, that is, f (i) �= i for all i.

{3, 1, 4, 2} is a derangement of {1, 2, 3, 4} but{3, 2, 4, 1} is not a derangement since 2 is left fixed.

What is the number of derangements , pn, of n elements?

12 CHAPTER 1. MATHEMATICAL REASONING, PROOF PRINCIPLES AND LOGIC

The number pn/n! can be interpreted as a probability .

Say n people go to a restaurant and they all check theircoat. Unfortunately, the cleck loses all the coat tags.Then, pn/n! is the probability that nobody gets her orhis coat back!

Interestingly, pn/n! has limit 1e ≈

13 as n goes to infinity,

a surprisingly large number.

1.1. PROBLEMS, MOTIVATIONS 13

1

16 17 18 19

10 11 12 13 14 15

5 6 7 8 9

1 2 3 4

Figure 1.3: An undirected graph modeling a city map

Problem 4. Finding strongly connected componentsin a directed graph.

The undirected graph of Figure 1.3 represents a map ofsome busy streets in a city.

The city decides to improve the traffic by making thesestreets one-way streets.

However, a good choice of orientation should allow oneto travel between any two locations. We say that theresulting directed graph is strongly connected .

14 CHAPTER 1. MATHEMATICAL REASONING, PROOF PRINCIPLES AND LOGIC

1

16 17 18 19

10 11 12 13 14 15

5 6 7 8 9

1 2 3 4

Figure 1.4: A choice of one-way streets

A possibility of orienting the streets is shown in Figure1.4.

Is the above graph strongly connected?

If not, how do we find its strongly connected compo-nents?

How do we use the strongly connected components to findan orientation that solves our problem?

1.1. PROBLEMS, MOTIVATIONS 15

1

d2

d3

dn+1

card 1card 2

card n

Figure 1.5: Stack of overhanging cards

Problem 5. The maximum overhang problem.

How do we stack n cards on the edge of a table, respectingthe law of gravity, and achieving a maximum overhang.

We assume each card is 2 units long.

Is it possible to achieve any desired amount of overhangor is there a fixed bound?

How many cards are needed to achieve an overhang of dunits?

16 CHAPTER 1. MATHEMATICAL REASONING, PROOF PRINCIPLES AND LOGIC

Problem 6. Ramsey Numbers

A version of Ramsey’s Theorem says that for every pair,(r, s), of positive natural numbers, there is a least positivenatural number, R(r, s), such that for every coloring ofthe edges of the complete (undirected) graph on R(r, s)vertices using the colors blue and red , either there is acomplete subgraph with r vertices whose edges are allblue or there is a complete subgraph with s vertices whoseedges are all red .

So, R(r, r), is the smallest number of vertices of a com-plete graph whose edges are colored either blue or redthat must contain a complete subgraph with r verticeswhose edges are all of the same color.

1.1. PROBLEMS, MOTIVATIONS 17

1

Figure 1.6: Left: A 2-coloring of K5 with no monochromatic K3; Right: A 2-coloring of K6

with several monochromatic K3’s

The graph shown in Figure 1.6 (left) is a complete graphon 5 vertices with a coloring of its edges so that there isno complete subgraph on 3 vertices whose edges are allof the same color.

Thus, R(3, 3) > 5.

There are215 = 32768

2-colored complete graphs on 6 vertices. One of thesegraphs is shown in Figure 1.6 (right).

It can be shown that all of them contain a triangle whoseedges have the same color, so R(3, 3) = 6.

18 CHAPTER 1. MATHEMATICAL REASONING, PROOF PRINCIPLES AND LOGIC

The numbers, R(r, s), are called Ramsey numbers .

It turns out that there are very few numbers r, s forwhich R(r, s) is known because the number of coloringsof a graph grows very fast! For example, there are

243×21 = 2903 > 102490 > 10270

2-colored complete graphs with 43 vertices, a huge num-ber!

In comparison, the universe is only approximately 14 bil-lions years old, namely 14× 109 years old.

For example, R(4, 4) = 18, R(4, 5) = 25, but R(5, 5) isunknown, although it can be shown that43 ≤ R(5, 5) ≤ 49.

Finding the R(r, s), or, at least some sharp bounds forthem, is an open problem.