Propositions and Boolean Formulæ

(命題と論理式)

Discrete Mathematics I

3rd lecture, October 4, 2019

https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture3.html

Martin J. Dürst

© 2005-19 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Summary of last lecture
• Last week's homework
• Propositions
• Construction and use of logical (Boolean) formulæ
• This week's homework

Summary of Last Lecture

• The history of numbers starts with the integers 1, 2, 3,...
• Across different cultures, the numerals for 1, 2, 3,... and 10, 20,... look similar
• Peano created a set of 5 axioms using "1" and "successor" only
• These axioms form a base for arithmetic on natural numbers
• The numeral 0 was discovered very late.
  It is important in positional number representation.
• Positional number representation can use various bases
• In Information Technology, the bases 2, (8,) 10, and 16 are used

Last Week's Homework: Jokes

Question: Why do computer scientist always think Christmas and Halloween are the same?

[Sorry, removed!]

Question: At what age do Information Technologists celebrate "Kanreki" (還暦)

64 = 2⁶ ("Kanreki" is a round number in the East Asian calendar. For Information Technology, powers of 2 are round numbers.)

Last Week's Homework: Arithmetic and Base Conversion

About Moodle

Those who have not yet registered for "Discrete Mathematics I" at http://moo.sw.it.aoyama.ac.jp, and those who have their name in Romaji instead of Kanji, must come to the front immediately after this lecture

[まだ http://moo.sw.it.aoyama.ac.jp で授業登録できてない人 、又は漢字ではなくローマ字で名前を登録している人は必ず授業直後申し出ること!]

How to Watch Videos

The video of the first two lectures are available via a link from Moodle.

Userid and password are the same as for other Aoyama websites.

Please use the video soon to review the lecture.
Please be careful when watching the video on a mobile device (may be expensive!).

The video can be watched at different speeds, and you can jump easily to the next slide.

Proposition

• A proposition is a sentence that is objectively either correct or wrong .
• Propositions may also be called statements.
• Even if a sentence is not correct, it is a proposition.
• Even if we do not know the answer (but there is a single answer), it is a proposition.
• Examples:
  • 2 times 2 is 4.
  • 10 is smaller than 5.
  • Four is an odd number.
  • On January 1, 2020, it will rain in Fuchinobe.

Counterexamples of Propositions

The following are NOT propositions:

• Subjective sentences (opinions,...)
• Questions
• Sentences that include undefined variables or pronouns

Counterexamples (NOT propositions):

• Her birthday is tomorrow (October 4, 2019).
  Reason: 'Her' is undefined.
• x is even.
  Reason: Undefined variable.
• Did I eat breakfast this morning?
  Reason: Question
• Natto (納豆) is delicious.
  Reason: Subjective opinion.

Truth Values of a Proposition

• Each proposition is either correct or wrong.
• "Correct" is called true (真), and is written T, ⊤, or 1.
• "Wrong" (not correct) is called false (偽), and is written F, ⊥, or 0.
• True and false together are called truth values .
• Other representations for truth values: ○/×, on/off, ...
• Truth values are also called Boolean values or just Booleans (after Georges Boole)

Truth Values in Program Languages

Different programming languages handle truth values differently.

Some examples:

• C programming language:
  • 0 represents false.
  • All integers ≠0 can represent true.
  • The type int is used for booleans.
  • The C99 standard introduced bool, true, and false.
• Unix shell:
  • 0 represents true.
  • All integers ≠0 can represent false.
  • This is exactly the opposite of C.
• Java programming language:
  • false and true are represented as false and true.
  • Integers (int) and boolean values (boolean) are different types.
• Ruby programming language:
  • false and true are represented as false and true.
  • false and nil (empty/nothing) are interpreted as false. All other objects are interpreted as true.

Logical Operations

With logical operations, we can create compound propositions from simpler propositions.

The most frequently used, and most basic, logical operations are:

• and: A ∧ B (conjunction)
• or: A ∨ B (disjunction)
• not: ¬A (negation)

Overview of Conjunction (and)

Based on two propositions A and B, we can construct the following proposition:

A and B

We write A ∧ B (also: A·B, A B, AND(A, B))

Because ∧ has two operands, it is a binary operation.

A ∧ B is defined as follows:
If both A and B are true, then A ∧ B is true.
Else, A ∧ B is false.

Truth Table for Logical And

Because there are only two truth values, we can define logical operations using a table. This is similar to the 9×9 multiplication table,

but would not work for operations on integers (e.g. multiplication) in general.

Examples for Logical And

7 is smaller than 5 ∧ 7 is even (read: 7 is smaller than 5 and is even): F

8 is smaller than 5 ∧ 8 is even (read: 8 is smaller than 5 and is even): F

4 is smaller than 5 ∧ 4 is even (read: 4 is smaller than 5 and is even): T

3 is smaller than 5 ∧ 3 is even (read: 3 is smaller than 5 and is even): F

Overview of Disjunction (or)

Based on two propositions A and B, we can construct the following proposition:

A or B

A or B is written A ∨ B (also: A+B, OR(A, B))

A ∨ B is defined as follows:
If both A and B are false, then A ∨ B is false.
Else, A ∨ B is true.

This means that logical or is inclusive. Logical or is a binary operation.

Truth Table for Logical Or

Examples for Logical Or

4 is smaller than 5 ∨ 4 is even: T

8 is smaller than 5 ∨ 8 is even: T

7 is smaller than 5 ∨ 7 is even: F

3 is smaller than 5 ∨ 3 is even: T

Overview of Negation (not)

Based on one proposition A, we can construct the following proposition:

not A

not A is written ¬A (also A', A, ~A, NOT(A)).

¬A is true if A is false

¬A is false if A is true.

Negation has only one operand and is therefore a unary operation.

Truth Table for Logical Not

Examples for Logical Not

¬ (4 is even) (read: 4 is not even): F
¬ (4 is odd) (read: 4 is not odd): T

The Importance of Logical Operations

In Information Technology:

• Create models of the real world
• Model logical operations in programming languages
• Model electronic circuits

In Mathematics:

• Model mathematical statements and their relationships
• Verify proofs
• Automate mathematical reasoning

Modeling

• Express facts of the real world as propositions
• Express relationships between facts as relationships between propositions
• Set up and work out logical formulæ using the propositions
• Apply the results to the real world

Examples: Conditions for getting a discount; conditions for a passing grade in a course; conditions for completing (winning) a game,...

Modeling Example

Condition for getting a discount: You get a discount if you are a student of Aoyama Gakuin University and today is a Friday or a Saturday.

Propositions:

Ao: You are a student of Aoyama Gakuin University.

Fr: Today (October 4, 2019) is a Friday.

Sa: Today (October 4, 2019) is a Saturday.

The truth values for these propositions are: Ao=T, Fr=T, Sa=F

The condition for a discount can be expressed as:
Ao ∧ (Fr∨Sa)

The condition evaluates to T, so you get a discount!

Structure of Logical Formulæ

We can creat a logical formula from propositions, propositional variables, and logical operators.

Example: (A ∨ (¬B)) ∧ C

Operator precedence and omission of parentheses:

• Each operator has a given precedence.
• Operations with higher precedence get evaluated first.
• If the order of evaluation is not correct, we use parentheses to fix it.

Precedence for logical operators:

• High: ¬
• Middle: ∧
• Low: ∨

Examples:
A∨B∧C = A ∨ (B∧C) ≠ (A∨B) ∧ C
(A ∨ (¬B)) ∧ C = (A ∨ ¬B) ∧ C ≠ A ∨ ¬B ∧ C

Well-Formed Formula

(WFF)

Goal: Make the structure (grammar) of logical formulæ clear.

All of the following are well-formed formulæ

• Propositions
• Propositional variables
• If W and V are well-formed formulæ, then the following are also well-formed formulæ:
  1. (W)
  2. ¬W
  3. W ∧ V
  4. W ∨ V

Formulæ that do not fit the above definition are not well-formed formulæ.
(Attention: In a later lecture, we will introduce more logical operators. This will extend the definition.)

Examples:
(A ∨ ¬B) ∧ C (well-formed)
(A ∨(B ¬∧) C (not well-formed)

Formulæ as Trees

• Formulæ can be drawn as trees
• Trees have a root (at the top), internal nodes, and leaf nodes
• The internal nodes are operators
• The leaves are operands
• The tree shows a formula's structure
• Evaluation order is expressed in the tree's structure (bottom to top)
  → parentheses are unnecessary

Example of Formula as a Tree

Example formula: A ∨ ¬B ∧ C

Logical Formula Evaluation with a Truth Table

Truth tables are often used to:

• Define logical operations
• Evaluate logical formulæ
• Prove properties of logical operations and formulæ [see lecture 5]

Example formula: (A ∨ ¬B) ∧ B

Evaluation using a truth table

From this truth table, we can deduce that (A ∨ ¬B) ∧ B = A ∧ B

How to Write a Truth Table

• Columns:
  • A column for each variable that appears in the formula
  • A column for each subexpression in the formula
    (starting with the the simplest subexpression)
  • A column for the overall formula
• Rows:
  • One row for the headers
    
  • 2ⁿ rows for entries, where n is the number of variables
• Variable columns:
  • Start with FF... in the first row
  • End with TT... in the last row
  • When converting F to 0 and T to 1, the variable columns should show the binary numbers from 0 to 2ⁿ-1.
• Subexpression columns and final column: Calculating each value from earlier columns.
• Example: Write a truth table for D ∧ ¬F ∨ ¬E

Boolean Functions

• A boolean function calculates a truth value from zero or more Boolean arguments.
• The domain of a boolean function are zero or more truth values.
• The range of a boolean function is a single truth value.
• Because Boolean arguments only take the values true and false, the number of Boolean functions is quite limited.

Number of Boolean Functions for n Variables

• Number of rows of a truth table with n variables: 2ⁿ
• Each row can have either T or F as a result
• Number of Boolean functions with n variables: 2⁽²ⁿ⁾
• Examples:

  Number of variables (n)	0	1	2	3	4	5
  Number of rows	1	2	4	8	16	32
  Number of Boolean functions	2	4	16	256	65536	4294967296

Question: Can we find a logic formula for any such function? (Answer: see next lecture)

Simple Laws for Arithmetic Operations

(for integers or reals,...)

• Commutative laws:
  Addition and multiplication are commutative:
  a+b =  b+a,   ab = ba
• Associative laws:
  Addition and multiplication are associative:
  (a+b) + c = a + (b+c),   (ab) · c = a ·  (b·C)
• Distributive law:
  Multiplication distributes over addition:
  a · (b+c) =  ab + ac
  Addition does not distribute over multiplication:
  a + bc ≠ (a+b) · (a+c)

Simple Laws for Logical Operations

• Commutative laws:
  Both conjunction and disjunction are commutative:
  A∧B = B∧ A,   A∨B = B∨A
• Associative laws:
  Both conjunction and disjunction are associative:
  (A∧B) ∧ C = A ∧ (B∧C),   (A∨B) ∨ C = A ∨  (B∨C)
• Distributive laws:
  Conjunction distributes over disjunction:
  A ∧ (B∨C) =  A∧B ∨ A∧C
  Disjunction distributes over conjunction:
  A ∨ B∧C =  (A∨B) ∧ (A∨C)

This Week's Homework

Repeat each quiz until you get 100% correct answers; deadline: October 10 (Thursday) 22:00:

1. Solve the quiz Propositions: True or False
2. Solve the quiz Truth Table 1
3. Solve the quiz Truth Table 2
4. Use highschool texts or the Web to research about other laws for logical operations.

今回の宿題

下記のクイズは満点まで繰り返す; 締切: 10月10日 (木) 22:00:

1. Moodle で Propositions: True or False のクイズを解く
2. Moodle で Truth Table 1 のクイズを解く
3. Moodle で Truth Table 2 のクイズを解く
4. 論理演算の (交換律、結合律、分配率以外の) 法則について高校の資料やウェブで調べること。

Glossary

round number

切りのいい数

proposition

命題

objective (adj.)

客観的

subjective (adj.)

主観的

opinion

意見

variable

変数

pronoun

代名詞

odd number

奇数

even number

偶数

truth value (of a proposition)

(命題の) 真偽

true

真 (しん)

false

偽 (ぎ)

type (in programming languages)

(プログラム言語の) 型

logical operation

論理演算

electronic circuits

電子回路

reasoning

推理、推論

modeling

モデル化 (する)

(logical) and

かつ

binary operation

二項演算

unary operation

単項演算

conjunction

論理積

truth table

真理表 (又は真理値表)

logical operation

論理演算

compound proposition

複合命題

(logical) or

又は

disjunction

論理和

(logical) not

ではない

negation

論理否定

logical formula

論理式 (複数: formulæ or formulas)

well-formed (logical) formula

整論理式

grammar

文法

tree

木 (生物学のものではなく、数学や情報テクノロジーで使うもの)

root

(木の) 根

internal node

内部節

leaf

葉

propositional variable

命題変数

(operator) precedence

(演算子の) 優先順位

evaluation

(式の) 評価

parentheses

括弧

subexpression

部分式

Boolean function

論理関数、ブール関数

Boolean argument

論理引数 (ひきすう; ブール引数とも言う)

domain (of a function)

(関数の) 定義域

range (of a function)

(関数の) 値域

commutative law

交換律

distributive law

分配律

distribute

分配する