Propositions and Boolean Formulæ

(命題と論理式)

Discrete Mathematics I

3rd lecture, October 3, 2014

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture3.html

Martin J. Dürst

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, 10 are similar
Peano created a set of 5 axioms using "1" and "successor" only that form a base for integer arithmetic
The numeral 0 was discovered very late, but 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?

[都合により削除]

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

[都合により削除]

Last Week's Homework: Arithmetic and Base Conversion

Total number of attempts: 452
Completely solved: 121 students
Less than 100%: 10 students
Not solved: 6 students
Average: 9.84 (out of 10)

About Moodle

Those who have not yet registered for "Discrete Mathematics I" at http://moo.sw.it.aoyama.ac.jp 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は授業直後申し出ること!]

Proposition

A proposition is a sentence that is objectively either correct or not.

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, 2015, 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, 2014).
Reason: 'Her' is undefined.
x is even.
Reason: Undefined variable.
When did you get up this morning?
Reason: Question
Natto (納豆) is delicious.
Reason: Subjective opinion.

Truth Values of a Proposition

Each proposition is either correct or not.

"Correct" is called true (真), and is written T, ⊤, or 1.

"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: ○/×,...

Truth Values in Program Languages

The handling of truth values (also called booleans) differs for different programming languages:

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

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

Logical Operations

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

The most frequently used logical operations are as follows:

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

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.

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

Truth Table for Logical And

`A`	`B`	`A` ∧ `B`
F	F	F
F	T	F
T	F	F
T	T	T

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.

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

Truth Table for Logical Or

`A`	`B`	`A` ∨ `B`
F	F	F
F	T	T
T	F	T
T	T	T

Negation

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 an unary operation.

Examples:
4 is not even. F
4 is not odd. T

Truth Table for Logical Not

`A`	¬`A`
F	T
T	F

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.
Operators with higher precedence get evaluated first.
If the order of evaluation is not correct, we use parentheses to fix it.

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

Example: 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:
- (W)
- ¬W
- W ∧ V
- 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.)

Logical Formula Evaluation with a Truth Table

Truth tables are often used for:

Defining logical operations
Evaluating logical formulæ
Proofs of properties of logical operations and formulæ

Example formula: (A ∨ (¬B)) ∧ B

Simplification (the parentheses around ¬B are unnecessary): (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

Create a column for each variable that appears in the formula.
Starting the the simplest subexpression, add a column for each subexpression in the formula.
Add a column for the overall formula.
Create one row for the headers, and 2ⁿ rows for entries, where n is the number of variables
Fill in the 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.
Fill in the subexpression columns and the final column by calculating each value from earlier columns.
Example: Write a truth table for (D ∧ ¬F) ∨ ¬E

How to Use a Truth Table for a Proof

Goal: Prove one of the absorbtion laws for ∨ and ∧: A∨A∧B = A.
Method:
- Create a truth table for both sides of the law.
- Compare the columns for both sides.
- If the columns are equal, the law is proved.

`A`	`B`	`A` ∧ `B`	`A` ∨ `A`∧`B`
F	F	F	F
F	T	F	F
T	F	F	T
T	T	T	T

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.

This Week's Homework

Solve the quiz Propositions: True or False until October 9, 22:00
Solve the quiz Truth Table 1 until October 9, 22:00
Solve the quiz Truth Table 2 until October 9, 22:00

Deadline: October 9, 22:00

Repeat each quiz until you get 100% correct answers.

今回の宿題

Moodle で Propositions: True or False のクイズを解く (全正解でない場合、繰り返す、締切: 10月 9日 (木) 22:00)
Moodle で Truth Table 1 のクイズを解く (全正解でない場合、繰り返す、締切: 10月 9日 (木) 22:00)
Moodle で Truth Table 2 のクイズを解く (全正解でない場合、繰り返す、締切: 10月 9日 (木) 22:00)

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
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
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