Predicate Logic and Quantifiers

(述語論理、全称限量子、存在限量子)

Discrete Mathematics I

7th lecture, November 15, 2024

https://www.sw.it.aoyama.ac.jp/2024/Math1/lecture7.html

Martin J. Dürst

© 2005-24 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Discussion of last week's test
• Leftovers/summary for last lecture
• Last week's homework
• Pascal's triangle, combinations, and combinatorics
• Predicates and predicate logic
• Operators on Predicates: Quantifiers
• This week's homework
• [after end of lecture: return of last week's test]

Summary of Last Lecture

• Sets are one of the most basic kinds of objects in Mathematics
• Sets are unordered, and do not contain any repetitions
• There are two notations for sets: Connotation (e.g. {1, 2, 3, 4}) and denotation (e.g. {n|n ∈ ℕ, n>0, n<5})
• Operations on sets include union, intersection, difference, and complement
• The empty set is a subset of any set
• The powerset of a set is the set of all subsets
• The laws for sets are very similar to the laws for predicate logic

Leftovers of Last Lecture

Neutral elements, Venn diagrams, laws for sets, limits of sets,...

Homework, Problems 1/2

1. Create a set with four elements. If you use the same elements as other students, a deduction of points will be applied.

Example: {cat, cow, crow, camel}

2. Create the powerset of the set you created in problem 1.

Example: {{}, {cat}, {cow}, {crow}, {camel}, {cat, cow}, {cat, crow}, {cat, camel}, {cow, crow}, {cow, camel}, {crow, camel}, {cat, cow, crow}, {cat, cow, camel}, {cat, crow, camel}, {cow, crow, camel}, {cat, cow, crow, camel}}

Homework, Problems 3/4

3. For sets A of size zero to six, create a table of the sizes of the powersets (|P(A)|).

4. Express the relationship between the size of a set A and the size of its powerset P(A) as a formula.

|P(A)| = 2^|A| (the size of the powerset of A is 2 to the power of the size of A)

Homework, Problem 5

5. Explain the reason behind the formula in problem 4.

The formula is correct for |A|=0: A={}, P(A)={{}}, |P(A)|=2⁰=1

If the formula is correct for |A|=k (i.e. |P(A)|=2^k),
we can show that it is correct for B with |B|=k+1

General case

B = {c|c∈A∨c=d∧d∉A} (= A∪{d} where d∉A),
|B| = k+1 = |A|+1,
P(B)=P(A)∪{e∪{d}|e∈P(A)},
|P(B)|=2·|P(A)| = 2·2^k=2^k+1

(This explanation is using Mathematical induction over k.)

Example

A={cat, cow}, |A|=2, P(A) = {{}, {cat}, {cow}, {cat, cow}}, |P(A)|=4

B = A∪{carp}={cat, cow, carp}, |B|=3

P(B) = P(A)∪{a∪{carp}|a∈P(A)} = P(A)∪{{carp}, {cat, carp}, {cow, carp}, {cat, cow, carp}}

|P(B)| = 2·|P(A)| = 8

Homework, Problem 6

6. Create a table that shows, for sets A of size zero to five, and for each n (size of sets in P(A)), the number of such sets.

(These numbers are the numbers appearing in Pascal's triangle.)

Pascal's Triangle

Start with a single 1 in the first row, surrounded by zeroes ((0 ... 0) 1 (0 ... 0)).
Create row by row by adding the number above and to the left and the number above and to the right.

                  1
                1   1
              1   2   1
            1   3   3   1
          1   4   6   4   1
        1   5  10  10   5   1
      1   6  15  20  15   6   1
    1   7  21  35  35  21   7   1
  1   8  28  56  70  56  28   8   1

Subsets and Pascal's Triangle

• For a set A with |A| = n, we can write
  |{B|B⊂A∧|B|=m}| as _nC_m
  (the mth number in the nth row of Pascal's triangle)
• _nC₀ = 1 (the only subset of size 0 is {}, {B|B⊂A∧|B|=1} = {{}})
• _nC_n = 1 (the only subset of size n is A itself, {B|B⊂A∧|B|=|A|=n} = {{A}})
• _nC_m = _n-1C_m-1 + _n-1C_m (n>0, 0<m<n)

Subsets and Combinations

• Combinatorics is very important for Information Technology
• Combinatorics deals with counting the number of different things under various conditions or restrictions
• The word combinations refers to the choices of a given size m from a set (of size n) without repetitions and without considering order
• The number of combinations is written _nC_m
• The number of combinations of size m selected from a set of size n are the same as the subsets of a given size m in a powerset of a set of size n (_nC_m = |{ S | S⊂A ∧ |S|=m}| where |A| = n)
• The number of combinations can be calculated directly: _nC_m = n!/(m! (n-m)!)
• There are also permutations (considering order), repeated permutations (allowing an element to be selected more than once), and repeated combinations

Types of Symbolic Logic

• Binary (Boolean) logic (using only true and false)
• Multi-valued logic (using e.g. true, false, and unknown)
• Fuzzy logic (including calculation of ambiguity)

• Propositional logic (using only propositions)
• Predicate logic (first order predicate logic,...)
• Temporal logic (integrating temporal relationships)

Limitations of Propositions

With propositions, related statements have to be made separately

Examples:
2 is even. 5 is even.
Today it is sunny. Tomorrow it is sunny. The day after tomorrow, it is sunny.

We can express "If today is sunny, then tomorrow will also be sunny." or "If 2 is even, then 3 is not even."

But we cannot express "If it's sunny on a given day, it's also sunny on the next day." or "If x is even, then x+2 is also even.".

⇒ This problem can be solved using predicates

Examples of Predicates

• even(4): 4 is even
• even(27): 27 is even
• odd(4): 4 is odd
• sunny(November 15, 2024): It is sunny on Noveber 15, 2024
• parent(Ieyasu, Hidetada): Ieyasu is the parent of Hidetada
  
• smaller(3, 7) (or: 3 < 7)

Predicate Overview

• The problem with propositions can be solved by introducing predicates.
• In the same way as propositions, predicates are objectively true or false.
• A predicate is a function (with 0 or more arguments) that returns true or false.
• If the value of an argument is undefined, the result (value) of the predicate is unknown.
• A predicate with 0 arguments is a proposition.

How to Write Predicates

There are two ways to write predicates:

1. Functional notation:
   • The name of the predicate is the name of the function
   • Arguments are enclosed in parentheses after the function name
   • Each predicate has a fixed number of arguments
   • Arguments in different positions have different meanings
   • Reading of predicates depends on their meaning
2. Operator notation (mainly for predicates with two arguments):
   • Operators that return true or false are predicates
   • Examples: 3 < 7, 5 ≧ 2, a ∈ B, even(x) ∨ odd(y)

Formulas Containing Predicates

Using predicates, we can express new things:

• sunny(x) → sunny(day after x)
• even(y) → even(y+2)
• even(z) → odd(z+1)

Similar to propositions, predicates can be true or false.

But predicates can also be unknown/undefined, for example if they contain variables.

Even if a predicate is undefined (e.g. even(x)),
a formula containing this predicate can have a defined value (true or false)
(e.g. even(y) → even(y+2), or odd(z) ↔ even(z+24))

First Order Predicate Logic

• The arguments of predicates can be constants, functions (e.g. sin), formulæ,...

  Examples:
  

  • even(2), say(Romeo, 'I love you'), parent(Ieyasu, Hidetada)
  • even(sin(0)), even(2+3×7)
• However, it is not possible to use predicates within predicates
  Counterexample: say(z, parent(y, x))
  (z says "y is the parent of x")
• Higher-order logic allows predicates within predicates

Universal Quantifier (∀)

Example: ∀n∈ℕ: even(n) → even(n+2)

Readings:

• For all n, elements of ℕ, if n is even, then n+2 is even.
• For all natural numbers n, if n is even, then n+2 is even.

General form: ∀x: P (x)

∀ is the A of "for All", inverted.

Readings in Japanese:

• 全ての自然数 n において、n が偶数ならば n+2 も偶数である
• 任意の x において、P(x)

Examples of Universal Quantifiers

∀n∈ℕ: n > -1

∀n∈ℕ: ∀m∈ℕ: n+m = m+n

∀a∈ℚ: ∀b∈ℕ: a+b = b+a

∀a∈{T, F}: ∀b∈{T, F}: a∨b = b∨a

Let S be the set of all students, B the set of all books, and let read(s, b) denote the fact that student s reads book b.

Then "∀s∈S: ∀b∈B: read(s, b)" means "all students read all books".

Remark 1: ∀n∈ℕ: ∀m∈ℕ: n+m = m+n can be written as ∀n, m∈ℕ: n+m = m+n

Remark 2: ∀s∈S: ∀b∈B: read(s, b) is interpreted as ∀s∈S: (∀b∈B: read(s, b))

Knowledge about Field of Application

• Propositional logic does not need application knowledge except for the truth value of each proposition.
• Predicate logic combines axioms/theorems/knowledge of logic with 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]

Existential Quantifier (∃)

Example: ∃n∈ℕ: odd(n)

Readings:

• There exists a n, element of ℕ, for which n is odd.
• There is a natural number n so that n is odd.
• There exists a natural number n which is odd.
• There exists an odd natural number.

General form: ∃y: P(y)

∃ is the mirrored form of the E in "there Exists".

Readings in Japanese:

• P(y) が成立する y が存在する
• ある y について、P(y)

Structure of Quantifier Expressions

Example: ∀m, n∈ℕ: m > n → m² ≧ n²

• ∀: Quantifier
• m, n: Variable(s), separated by commas
• ∈ℕ: Set membership (applies to all previous variables connected by commas; unnecessary if there is a single obvious universal set)
• ":": Colon
• m > n → m² ≧ n²: Quantified predicate

More Quantifier Examples

∀n∈ℕ: n + n + n = 3n

∃n∈ℕ: n² = n³

∃q∈ℝ: q² < 50q < q³

∀m, n∈ℕ: 7m + 2n = 2n + 7m

Applied Quantifier Examples

S: Set of students

F: Set of foods

like(s, f): Student s likes food f

1. All students like all foods:
2. Some students like all foods:
3. There is a food that all students like:
4. There is no food that all students like:
5. Each student dislikes a food:

Summary

• Predicates are limited because they cannot express complex facts
• Propositions are functions or operators that return truth values (true/false)
• First order propositional logic allows formal reasoning for some application domain (e.g. arithmetic, set theory,...)
• The universal quantifier (∀) expresses that a predicate is true for all members of some set(s)
• The existential quantifier (∃) expresses that a predicate is true for some member(s) of some set(s)

Homework Due November 21

Deadline: November 21, 2024 (Thursday), 18:30.

Format: Handout, easily readable handwriting

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

Problems: See handout

About Returns of Tests and Homeworks

• Today, the graded intermediate tests will be returned
• This is not part of the lecture itself (i.e. after 12:30)
• If you have some other commitment after 12:30, you can come to my office next week to pick up your homework
• Tests will be distributed in order of student number (increasing). Please make sure you are at the front of the lecture hall to get your test when your number/name is called; otherwise you have to wait for the second round.
• When taking your homework, make sure it is really yours
• NEVER take the homework of somebody else (a friend,...)
• Carefully analyze your mistakes and unanswered problems and work on improving to avoiding similar mistakes and omissions in the future
• Feel free to ask questions, preferably on the Moodle Q&A forum

Glossary

Mathematical induction

数学的帰納法

Pascal's triangle

パスカルの三角形

combinatorics

組合せ論

combination

組合せ

permutation

順列

repeated combination

重複組合せ

repeated permutation

重複順列

predicate logic

述語論理

quantifier

限量子

evaluate

評価する

evaluation

評価

array

配列

symbolic logic

記号論理

multi-valued logic

多値論理

fuzzy logic

ファジィ論理

ambiguity

曖昧さ

first-order predicate logic

一階述語論理

temporal logic

時相論理

binary logic

二値論理

generalization

一般化

argument

引数

undefined

未定

higher-order logic

高階述語論理

universal quantifier

全称限量子 (全称記号)

existential quantifier

存在限量子 (存在記号)