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

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:

Reductio ad absurdum: A→¬A = ¬A
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Contraposition: A→B = ¬B→¬A
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The associative law for disjunction: (A∨B) ∨ C = A ∨ (B∨C)
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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