# Applications of Predicate Logic

(述語論理の応用)

## Discrete Mathematics I

### 7th lecture, Nov. 6, 2015

http://www.sw.it.aoyama.ac.jp/2015/Math1/lecture7.html

### Martin J. Dürst # Today's Schedule

• Summary and homework from 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 (∃)

# 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 for 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 5, Problem 3

Law 5:
(∃x: P(x)) ∧ Q(y) = ∃x: (P(x)∧Q(y))

Concrete example:

There is a student from Shizuoka, and student y is older than 20 means that there is a case where there is a student from Saitama and student y is older than 20, and the other way round.

(∃s: native(s, Shizuoka)) ∧ age(y)>20 = ∃s: (native(s, Shizuoka)∧age(y)>20)

# Homework Due November 5, Problem 3

Law 8:
(∀x: P(x)) ∧ (∀x: R(x)) = ∀x: P(x)∧R(x)

Concrete example:
All students are less that 30 years old, and all students belong to the College of Science and Engineering. = All students are less than 30 years old and belong to the College of Science and Engineering.

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

# Homework Due November 5, Problem 3

Law 11:
(∃x: P(x)) ∧ (∃x: R(x)) ← ∃x: P(x) ∧ R(x)

Example of how the left side follows from the right:
There is a student who is a native of Hiroshima and who is male. → There is a student who is a native of Hiroshima, and there is a student who is male.

(∃s: native(s, Hiroshima)) ∧ (∃s: male(s)) ← ∃s: (native(s, Hiroshima) ∧ male(s))

Example of how the right side does not follow from the left side:
There is a student who is a native from Hokkaido, and there is a student who is female. However, this does not imply that there is a student who is a native from Hokkaido and is female.

(∃s: native(s, Hokkaido)) ∧ (∃s: female(s)) ↛ ∃s: (native(s, Hokkaido) ∧ female(s))

# Homework Due November 5, Problem 3

Law 12:
(∃y: ∀x: P(x, y)) → (∀x: ∃y: P(x, y))

Example of how the left side follows from the right:
There is an age y (e.g. 30) so that for all students, their age is below y. From this follows that for each student, there is an age for which the student's age is lower.

(∃y: ∀s: age(s)<y) → (∀s: ∃y: age(s)<y)

Example of how the right side doesn't follow from the left side:
All students are native of some prefecture (or abroad). But this does not mean that there is a single prefecture of which all students are native.

(∀s: ∃y: native(s, y)) ↛ (∃y: ∀s: native(s, y))

# 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) → yz
• Calculate a new number t = 1 + zp=2(prime(p)) p
Example: z=7; t = 1 + 2·3·5·7 = 211
• xz: prime(x)→t mod x =1 ⇒

prime(t) ∨ ∃s: (z<s<t ∧ prime(s)) ⇒

y: (y > x ∧ prime(y))

# 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)=CSE) ⇔ ∀u: (age(u)≤30 ∧ college(u)=CSE)
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)=CSE) = ∀u: (age(u)≤30∧college(u)=CSE

is the same as

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

There are three different variables s in the last statement.

Advice:

• It is better to use different variable names with each quantifier, but
• You have to be able to handle formuæ that repeatedly used the same variable name, too.

# 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)∨...

Quantification is a generalization of conjunction/disjunction to more than two operands in the same way that sum and product are a generalization of addition/multiplication to more than two operands.

# Quantifiers for Empty Sets

i (i<0⋀i>5): odd(i) = T

i (i<0⋀i>5): odd(i) = F

# Extension of DeMorgan's Laws

Laws 1 and 2 introduced in the last lecture are generalizations 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)∧...)

# Formula Manipulation with Quantifiers

Simplify ¬(∃x: P(x) → ∀y: ¬Q(y))

¬(∃x: P(x) → ∀y: ¬Q(y)) [removing implication]

= ¬(¬∃x: P(x) ∨ ∀y: ¬Q(y)) [deMorgan's law]

= ¬¬∃x: P(x) ∧ ¬∀y: ¬Q(y) [law 1 of last lecture]

= ¬¬∃x: P(x) ∧ ∃y: ¬¬Q(y) [double negation]

= ∃x: P(x) ∧ ∃y: Q(y)

Actual example:

Let P(x) mean "it is raining in x", and Q(y) "it is snowing y"

Then the original formula says "It's wrong that if it rains somewhere, then it snows nowhere". The final formula says "There is a place where it rains and there is a place where it snows".

# 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)) [all students are either male or female]

s: ¬(male(s) ∧ female(s)) [no student is both male and female]

sS: (∃kK: native(s, k) ∧(∀hK: h=k ∨¬native(s, h))) [all students are native of exactly one prefecture]

# This Week's Homework

Deadline: November 12, 2015 (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

native of ...
...出身
bound variable

free variable

closed formula

scope

sum

product

prime number

infinite

set

element

(set) union

(set) intersection

(set) difference

subset

proper subset

empty set

power set
べき (冪) 集合