Discrete Mathematics: Proof Techniques and Number Theory, Study notes of Discrete Mathematics

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Discrete Mathematics

(Proof Techniques)

Pramod Ganapathi

Department of Computer Science State University of New York at Stony Brook

March 21, 2022

What is a proof?

Definition A proof is a method for establishing the truth of a statement. Rigor Truth type Field Truth teller 0 Word of God Religion God/Priests 1 Authoritative truth Business/School Boss/Teacher 2 Legal truth Judiciary Law/Judge/Law makers 3 Philosophical truth Philosophy Plausible argument 4 Scientific truth Physical sciences Experiments/Observations 5 Statistical truth Statistics Data sampling 6 Mathematical truth Mathematics Logical deduction

Why care for mathematical proofs?

The current world ceases to function without math proofs (My belief) Reduction tree showing subjects that possibly could be expressed or understood in terms of other subjects Humanities

Psychology

Biology

Chemistry

Physics

Mathematics CS

Methods of mathematical proof

Statements Method of proof Proving existential statements Constructive proof (Disproving universal statements) Non-constructive proof Proving universal statements Direct proof (Disproving existential statements) Proof by mathematical induction Well-ordering principle Proof by exhaustion Proof by cases Proof by contradiction Proof by contraposition Computer-aided proofs

Even and odd numbers

Definitions An integer n is even iff n equals twice some integer; Formally, for any integer n ,

n is even ⇔ n = 2 k for some integer k

An integer n is odd iff n equals twice some integer plus 1; Formally, for any integer n ,

n is odd ⇔ n = 2 k + 1 for some integer k

Examples Even numbers: 0 , 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 , 24 , 26 , 28 , 30 , 32 ,... Odd numbers: 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 25 , 27 , 29 , 31 , 33 , 35 ,...

Rational and irrational numbers

Definitions A real number r is rational iff it can be expressed as a ratio of two integers with a nonzero denominator; Formally, if r is a real number, then

r is rational ⇔ ∃ integers a, b such that r =

a b

and b 6 = 0

A real number r is irrational iff it is not rational Examples Rational numbers: 10 , − 56_._ 47 , 10 / 13 , 0 , − 17 / 9 , 0_._ 121212_... ,_ − 91 ,... Irrational numbers: √ 2 ,

√ 2 , π, φ, e, π^2 , e^2 , 21 /^3 , log 2 3 ,... Open problems: It’s not known if π + e, πe, π/e, πe, π

√ (^2) , and ln π are irrational

Quotient-Remainder theorem

Theorem Given any integer n and a positive integer d , there exists an integer q and a whole number r such that

n = qd + r and r ∈ [0 , d − 1]

Examples Let n = 6 and d ∈ [1 , 7] Num. ( n ) Divisor ( d ) Theorem Quotient ( q ) Rem. ( r ) 6 1 6 = 6 × 1 + 0 6 0 6 2 6 = 3 × 2 + 0 3 0 6 3 6 = 2 × 3 + 0 2 0 6 4 6 = 1 × 4 + 2 1 2 6 5 6 = 1 × 5 + 1 1 1 6 6 6 = 1 × 6 + 0 1 0 6 7 6 = 0 × 7 + 6 0 6

Prime numbers

• 2 2 = 1 × 2 = 2 × Num. Factorization Prime?
• 3 3 = 1 × 3 = 3 ×
• 4 4 = 1 × 4 = 4 × 1 = 2 ×
• 5 5 = 1 × 5 = 5 ×
• 6 6 = 1 × 6 = 6 × 1 = 2 × 3 = 3 ×
• 7 7 = 1 × 7 = 7 ×
• 8 8 = 1 × 8 = 8 × 1 = 2 × 4 = 4 ×
• 9 9 = 1 × 9 = 9 × 1 = 3 ×
• 10 10 = 1 × 10 = 10 × 1 = 2 × 5 = 5 ×
• 11 11 = 1 × 11 = 11 ×
• 12 12 = 1 × 12 = 12 × 1 = 2 × 6 = 6 × 2 = 3 × 4 = 4 ×
• 13 13 = 1 × 13 = 13 ×
• 14 14 = 1 × 14 = 14 × 1 = 2 × 7 = 7 ×
• 15 15 = 1 × 15 = 15 × 1 = 3 × 5 = 5 ×
• 16 16 = 1 × 16 = 16 × 1 = 2 × 8 = 8 × 2 = 4 ×
• 17 17 = 1 × 17 = 17 ×

Prime numbers

Definitions A natural number n is prime iff n > 1 and for all natural numbers r and s , if n = rs , then either r or s equals n ; Formally, for each natural number n with n > 1 ,

n is prime ⇔ ∀ natural numbers r and s , if n = rs then n = r or n = s

A natural number n is composite iff n > 1 and n = rs for some natural numbers r and s with 1 < r < n and 1 < s < n ; Formally, for each natural number n with n > 1 ,

n is composite ⇔ ∃ natural numbers r and s , if n = rs and 1 < r < n and 1 < s < n

Unique prime factorization of natural numbers

n Unique prime factorization 2 2 3 3 4 22 5 5 6 2 × 3 7 7 8 23 9 32 10 2 × 5 11 11 12 22 × 3 13 13 14 2 × 7 15 3 × 5

n Unique prime factorization 16 24 17 17 18 2 × 32 19 19 20 22 × 5 21 3 × 7 22 2 × 11 23 23 24 23 × 3 25 52 26 2 × 13 27 33 28 22 × 7 29 29

n Unique prime factorization 30 2 × 3 × 5 31 31 32 25 33 3 × 11 34 2 × 17 35 5 × 7 36 22 × 32 37 37 38 2 × 19 39 3 × 13 40 23 × 5 41 41 42 2 × 3 × 7 43 43

What is the pattern?

Some terms

Definitions Absolute value of real number x , denoted by | x | is

| x | =

{ x if x ≥ 0 − x if x < 0 Triangle inequality. For all real numbers x and y , | x + y | ≤ | x | + | y | Floor of a real number x , denoted by b x c is b x c = unique integer n such that n ≤ x < n + 1 b x c = n ⇔ n ≤ x < n + 1 Ceiling of a real number x , denoted by d x e is d x e = unique integer n such that n − 1 < x ≤ n d x e = n ⇔ n − 1 < x ≤ n

Some terms

Definitions Given an integer n and a natural number d , n div d = integer quotient obtained when n is divided by d , n mod d = whole number remainder obtained when n is divided by d. Symbolically, n div d = q and n mod d = r ⇔ n = dq + r where q and r are integers and 0 ≤ r < d.

Direct Proof

Even + odd = odd

Proposition Sum of an even integer and an odd integer is odd.