Propositions and Boolean Formulæ

(命題と論理式)

Discrete Mathematics I

3rd lecture, September 29, 2017

http://www.sw.it.aoyama.ac.jp/2017/Math1/lecture3.html

Martin J. Dürst

AGU

© 2005-17 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

 

Summary of Last Lecture

 

About Moodle

Those who have not yet registered for "Discrete Mathematics I" at http://moo.sw.it.aoyama.ac.jp, or 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!]

 

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:

 

Counterexamples of Propositions

The following are NOT propositions:

Counterexamples (NOT propositions):

 

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 in Program Languages

The handling of truth values (also called Boolean values or simply Booleans) differs among programming languages.

"Boolean" is named after Georges Boole.

Some examples:

 

Logical Operations

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

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

 

Conjunction (and)

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

A and B

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

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

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

Examples:
4 is smaller than 5 ∧ 4 is even. T (read: 4 is smaller than 5 and is even)
7 is smaller than 5 ∧ 7 is even. F (read: 7 is smaller than 5 and is even)
8 is smaller than 5 ∧ 8 is even. F (read: 8 is smaller than 5 and is even)
3 is smaller than 5 ∧ 3 is even. F (read: 3 is smaller than 5 and is even)

 

Truth Table for Logical And

A B AB
F F F
F T F
T F F
T T T

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.

 

Disjunction (or)

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

A or B

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

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

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

Examples:
4 is smaller than 5 ∨ 4 is even. T
8 is smaller than 5 ∨ 8 is even. T
3 is smaller than 5 ∨ 3 is even. T
7 is smaller than 5 ∨ 7 is even. F

 

Truth Table for Logical Or

A B AB
F F F
F T T
T F T
T T T

 

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, and is false if A is true.

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

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

Truth Table for logical not

A ¬A
F T
T F

 

The Importance of Logical Operations

In Information Technology:

In Mathematics:

 

Modeling

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 (2017/9/29) is a Friday.

Sa: Today (2017/9/29) 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 ∧ (FrSa)

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:

For logical operators, ¬ has higher precedence than ∧, which has higher precedence than ∨.

Examples:
ABC = A ∨ (BC) ≠ (AB) ∧ C
(A ∨ (¬B)) ∧ C = (A ∨ ¬B) ∧ CA ∨ ¬BC

 

Well-Formed Formula

(WFF)

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

All of the following are well-formed formulæ

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); (AB ¬∧) C (not well-formed)

 

Formulæ as Trees

 

Example of Formula as a Tree

Example formula: A¬BC

Logical Formula Evaluation with a Truth Table

Truth tables are often used to:

Example formula: (A ∨ ¬B) ∧ B

Evaluation using a truth table

A B ¬B A ∨ ¬B (A ∨ ¬B) ∧ B
F F T T F
F T F F F
T F T T F
T T F T T

 

How to Write a Truth Table

 
 
 
 
 
 
 

 

Boolean Functions

 

This Week's Homework

Repeat each quiz until you get 100% correct answers; deadline: October 5 (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 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 5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. 論理演算の法則 (例: 結合率) 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げ使うもの)
root
(木の) 根
internal node
内部節
leaf
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