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Logic and Discrete Mathematics

Data Science and Engineering

Marc Noy Anna de Mier, Juanjo Ru´e

Contents of the course

1. Sets and induction
2. Logic I (^) Proposicional calculus I (^) First order logic (predicate calculus)
3. Integers and polynomials I (^) Divisibility of integers I (^) Congruences I (^) Polynomials
4. Combinatorics I (^) Basic counting I (^) Permutations and cycles I (^) Recurrences
5. Graphs: theory and algorithms I (^) Graphs and digraphs I (^) Trees I (^) Minimum Spanning Tree and Travelling Salesman Problem I (^) Planar graphs and graph coloring

Basic concepts on sets

A set contains elements a ∈ S means element a is contained in set S Examples 2 ∈ N, 23 ∈/ Z,

2 ∈/ Q, 5 ∈ A, 5 ∈/ B

Defining sets A = { 2 , 3 , 5 , 6 } B = { 1 , 3 , 6 , 8 , 9 } W = {Mon, Tues, Wednes, Thus, Fri, Sat, Sun} E = {a ∈ Z|a = 2b for some b} E = {a ∈ Z : a = 2b for some b} I = {x ∈ R : a ≤ x ≤ b} I ′^ = {x ∈ R : a < x ≤ b}

The empty set ∅ contains no element A ⊆ B means A is a subset of B

Operations on sets

Union A ∪ B = {x : x ∈ A or x ∈ B} Intersection A ∩ B = {x : x ∈ A and x ∈ B} Complement A\B = A − B = {x : x ∈ A and x ∈/ B}

If all sets we consider are subsets of a fixed set U, define Ac^ = {x ∈ U : x ∈/ A} The following rules hold

1. A ∪ B = B ∪ A, A ∪ (B ∪ C ) = (A ∪ B) ∪ C = A ∪ B ∪ C
2. A ∩ B = B ∩ A, A ∩ (B ∩ C ) = (A ∩ B) ∩ C = A ∩ B ∩ C
3. A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )
4. A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
5. (A ∪ B)c^ = Ac^ ∩ Bc
6. (A ∩ B)c^ = Ac^ ∪ Bc The product A × B is the set of all ordered pairs (a, b): A × B = {(a, b) : a ∈ A, b ∈ B}

Induction proofs

Example We want to prove the formula

E (n) 12 + 2^2 + 3^2 + · · · + n^2 = n(n + 1)(2n + 1) 6 We first check that it is true for n = 1 We then check that if it is true for n, then it is also true for n + 2 That is E (n) =⇒ E (n + 1) Assuming E (n) true, we need to prove

E (n+1) 12 +2^2 +3^2 +· · ·+n^2 +(n+1)^2 = (n + 1)(n + 2)(2n + 3) 6 By the induction hypothesis we have

12 + 2^2 + 3^2 + · · · + n^2 + (n + 1)^2 =

n(n + 1)(2n + 1) 6

• (n + 1)^2 Now it is easy to check that the right-hand term is indeed equal to (n + 1)(n + 2)(2n + 3) 6 This concludes the proof

The induction principle

We want to prove a property P(n) for all integers n > 0 We must prove two things: I (^) P(1) is true I (^) For all n > 0, if P(n) is true then P(n + 1) is true The induction principle says that then P(n) is true for all n > 0 Induction can start at n = 0 or at any fixed positive integer. Examples I ∏ni=

1 − (^) i^12

= n 2 +1n. I (^) n! > 2 n^ for n ≥ 4. Here the induction starts at n = 4. I (^) For a real number x > −1, we have (1 + x)n^ ≥ 1 + nx for n ≥ 0. I (^) The Fibonacci numbers fn are defined by fn = fn− 1 + fn− 2 for n ≥ 2 , f 0 = 0, f 1 = 1. Then fn+1fn− 1 − f (^) n^2 = (−1)n, for n ≥ 1

1. LOGIC

1.1 Proposicional calculus

Atomic formulas are basic statements I (^) Today is Monday I (^) Anna has two children I (^) Wise people don’t smoke I (^) Robert is sick I (^) The party is tomorrow

We wish to combine these atomic statements to form ’formulas’

George Boole (1815–1864)

An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities

Assigning truth values (also known as interpretations)

Assign to each atom a truth value

t : {Atoms} → { 0 , 1 }

1 means True and 0 means False To each formula F we evaluate its truth value as follows I (^) If F = p is an atom, its value is t(p) I (^) t(¬F ) = 1 − t(F ) (true if F is false) I (^) t(F ∨ G ) = max{t(F ), t(G )} (true if at least one is true) I (^) t(F ∧ G ) = min{t(F ), t(G )} (true if both are true)

F ¬F

F G F ∧ G F ∨ G

Logical equivalence

Two formulas are logically equivalent if they give the same evaluation on all truth values of their arguments I (^) (p → q) ≡ (¬p ∨ q) I (^) (p ↔ q) ≡ (p → q ∧ q → p) Further equivalences I (^) F ∨ G ≡ G ∨ F (commutative) I (^) F ∨ (G ∨ H) ≡ (F ∨ G ) ∨ H (associative) I (^) F ∧ G ≡ G ∧ F I (^) F ∧ (G ∧ H) ≡ (F ∧ G ) ∧ H I (^) ¬¬F ≡ F I (^) ¬(F ∧ G ) ≡ ¬F ∨ ¬G (De Morgan) I (^) ¬(F ∨ G ) ≡ ¬F ∧ ¬G I (^) F ∧ (G ∨ H) ≡ (F ∧ G ) ∨ (F ∧ H) (Distributive) I (^) F ∨ (G ∧ H) ≡ (F ∨ G ) ∧ (F ∨ H) I (^) F ⊕ G ≡ (F ∧ ¬G ) ∨ (¬F ∧ G )

F is a Tautology if it always evaluates to 1 F is a Contradiction if it always evaluate to 0

p ∨ ¬p is a tautology p ∧ ¬p is a contradiction

If T is a tautology and C is a contradiction then

I (^) A ∨ T ≡ T I (^) A ∧ T ≡ A I (^) A ∨ C ≡ A I (^) A ∧ C ≡ C

Distinction between ↔ (connector) and ≡ (logical equivalence)

Procedure for transforming a formula into CNF

I (^) Eliminate → using A → B ≡ ¬A ∨ B I (^) Push negations down using I (^) ¬¬A ≡ A I (^) ¬(A ∧ B) ≡ ¬A ∨ ¬B I (^) ¬(A ∨ B) ≡ ¬A ∧ ¬B I (^) Use A ∨ (B 1 ∧ · · · ∧ Br ) ≡ (A ∨ B 1 ) ∧ · · · ∧ (A ∨ Br )

Logical consequence

Formula B is a logical consequence of A if

For every truth assignment of the literals, whenever A evaluates to 1, B also evaluates to 1

We write A |= B B is a logical consequence of {A 1 ,... , Am} if

A 1 ∧ · · · ∧ Am |= B