Applications of Predicate Logic

(述語論理の応用)

Discrete Mathematics I

7th lecture, Nov. 14, 2014

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture7.html

Martin J. Dürst

© 2005-14 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Summary and homework for last lecture
• Applications of predicate logic
  • Examples for various laws
  • Quantifiers and variables
  • Quantifiers and sums/products

Summary of Last Lecture

• There are many different kinds of symbolic logic: Propositional logic, predicate logic,...
• Predicates take arguments (propositions don't take arguments)
• Predicate logic allows more general inferences than propositional logic
• Predicate logic uses universal quantifiers (∀) and existential quantifiers (∃)

Homework Due October 30, Problem 1

Prove/check the following laws using truth tables:

1. Reductio ad absurdum: A→¬A = ¬A

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2. Contraposition: A→B = ¬B→¬A

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3. The associative law for disjunction: (A∨B) ∨ C = A ∨  (B∨C)

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4. One of De Morgan's laws: ¬(A∧B) = ¬A ∨ ¬B

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Homework Due October 30, Problem 2

Prove transitivity of implication (((A→B) ∧ (B→C)) ⇒ (A→C)) by formula manipulation.
Hint: Show that ((A→B) ∧ (B→C)) → (A→C) is a tautology by simplifying it to T.

For each simplification step, indicate which law you used.

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Homework Due November 13, Problem 1

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Homework Due November 13, Problem 1

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Homework Due November 13, Problem 2

For ternary (three-valued) logic, create truth tables for conjunction, disjunction, and negation. The three values are T, F, and ?, where ? stands for "maybe true, maybe false, we don't know".

Hint: What's the result of "?∨T"? ? can be T or F, but in both cases the result will be T, so ?∨T=T.

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Homework Due November 13, Problem 3

For each of the laws 8-11 of "Laws for Quantifiers", imagine a concrete example and explain it. For laws 9 and 11, give examples for both why the implication works one way and why the implication does not work the other way.

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Notation used in Examples

The examples all are about the students taking Discrete Mathematics I.

Predicates and functions used:

age(s): A student's age (fully completed years)

college(s): A student's faculty or college (example: College of Science and Engineering)

female(s), male(s): Predicates to test a student's gender.

native(s, k): True if student s is a native of prefecture k (using "abroad" for students from outside Japan)

Homework Due November 13, Problem 3

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Homework Due November 13, Problem 3

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Homework Due November 13, Problem 3

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The use of Variables with Quantifiers

Bound variable:

Variable quantified by a quantifier
Example: the x in: (∀x: P(x)∧Q(y))

Free variable:

Variable not quantified by a quantifier
Example: the y in: (∀x: P(x)∧Q(y))

Closed formula:

A formula without free variables.

Scope:

The part of a formula where a bound variable (or a quantifier) is active.
All occurrences of a bound variable within its scope can be exchanged by another variable.
Example: ∀s: (age(s)≤30 ∧ college(s)=SF) ⇒ ∀u: (age(u)≤30 ∧ college(u)=SF)
Using a bound variable outside its scope is an error.
Example: (∀x: P(x))∧Q(x)

Manipulation of Bound Variables

∀s: age(s)≤30) ∧ (∀t: college(t)=SF) = ∀u: (age(u)≤30 ∧ college(u)=SF

is the same as

∀s: age(s)≤30) ∧ (∀s: college(s)=SF) = ∀s: (age(s)≤30 ∧ college(s)=SF

There are three different variables s in the last statement.

Relation between Sums/Products and Quantifiers

Sum: ∑^∞_i=1 1/i² = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...

Product: ∏^∞_i=1 1+1/(-2)ⁱ = ...

Universal quantification: (∀i ∈ℕ⁺: i>0) = ⋀^∞_i=1 i>0 = 1>0 ∧2>0 ∧3>0 ∧...

Existential quantification: (∃i ∈ℕ⁺: odd(i)) = ⋁^∞_i=1 odd(i) = odd(1)∨odd(2)∨odd(3)∨...

Extension of DeMorgan's Laws

• ¬∀x: P(x) = ∃x: ¬P(x)
  ¬(P(x₁)∧P(x₂)∧P(x₃)∧P(x₄)∧...) =
  = (¬P(x₁)∨¬P(x₂)∨¬P(x₃)∨¬P(x₄)∨...)
• ¬∃x: P(x) = ∀x: ¬P(x)
  ¬(P(x₁)∨P(x₂)∨P(x₃)∨P(x₄)∨...) =
  = (¬P(x₁)∧¬P(x₂)∧¬P(x₃)∧¬P(x₄)∧...)

Combination of Quantifiers

(∃y: ∀x: P(x, y)) → (∀x: ∃y: P(x, y))

(∀x: ∃y: P(x, y)) ↛ (∃y: ∀x: P(x, y))

The number of prime numbers is infinite.

(This means that whatever big number x we choose, there will always be a bigger prime number y.)

∀x: ∃y: (y > x ∧ prime(y))

Reversing the order of the quantifiers changes the meaning:

∃y: ∀x: (y > x ∧ prime(y))

(There is a prime number y that is bigger than any (natural number) x. This statement is obviously false.)

Proof that the Number of Prime Numbers is Infinite

• Assume there is a largest prime number z:
  ∃z: prime(z) ∧ ∀y: prime(y) → y≤z
• Calculate a new number t = 1 + ∏^z_p=2(prime(p)) p
  Example: z=7; t = 2·3·5·7 + 1 = 211
• ∀x<z: prime(x)→t mod x =1 ⇒

  prime(t) ∨ ∃s: (z<s<t ∧ prime(s)) ⇒

  ∃y: (y > x ∧ prime(y))

Knowledge about Field of Application

• Propositional logic does not nead application knowledge except for the truth value of each proposition.
• Predicate logic combines axioms/theorems/knowledge of logic with the axioms/theorems/knowledge of one or more application areas.
• Example: Predicate logic on natural numbers: Peano axioms,...
• Example: Predicate logic for sets: Laws for operations on sets,...
• Example: Size of sets: Knowledge about set operations and arithmetic with natural numbers
• Concrete example:

  ∀s: (male(s) ∨ female(s))

  ∀s: ¬(male(s) ∧ female(s))

  ∀s∈S: (∃k∈K: from(s, k) ∧(∀h∈K: h=k ∨¬from(s, h)))

This Week's Homework

Deadline: November 20, 2014 (Thursday), 19:00.

Format: A4 single page (using both sides is okay; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Using your high school books/materials or other sources, research the following terms related to sets, and write a definition and short explanation for each of them:

• Set
• Element
• Set union
• Set intersection
• Set difference
• Subset
• Proper subset
• Empty set
• Power set

(高校の本・資料や他の情報を活用して、上記の集合に関連する概念を調査し、定義と簡単な説明を書きなさい。)

Glossary

College of Science and Engineering

理工学部

native of ...

...出身

bound variable

束縛変数

free variable

自由変数

closed formula

閉論理式

scope

作用領域、スコープ

sum

総和

product

総積

prime number

素数

infinite

無限 (な)

set

集合

element

元、要素

(set) union

和集合

(set) intersection

積集合

(set) difference

差集合

subset

部分集合

proper subset

真 (しん) の部分集合

empty set

空 (くう) 集合

power set

べき (冪) 集合