Boolean Algebras

(ブール代数)

Discrete Mathematics I

12th lecture, December 8, 2023

https://www.sw.it.aoyama.ac.jp/2023/Math1/lecture12.html

Martin J. Dürst

Today's Schedule

Summary, leftovers, and homework for last lecture
Algebraic structures related to groups
Boolean algebra
Examples of Boolean algebras
Bitstrings and Bitwise Operations
Structure of Boolean algebras

Summary of Last Lecture

Algebraic structures are defined as sets with operations and axioms
Groups are the best known example of an algebraic structure, with a rich theory
Groups have one associtive binary operation, an identity element, and inverse elements
Examples of groups are (ℤ, +), (ℝ-{0}, ·), (ℝ⁺, ·), and symmetric groups (permutations of n elements with composition)
Finite groups are represented by Cayley tables, and compared using the concept of isomorphism (one-to-one mapping conserving structure)

Leftovers from Last Lecture

Group theorems: Uniqueness of Inverse, Cancellation Law

Groups in Computer Science

Last Week's Homework 1:
Symmetric Group of Order 3

Create a Cayley table of the symmetric group of order 3. Use lexical order for the permutations.

[都合により削除]

Last Week's Homerwork 2:
Non-Isomorphic Groups of Size 4

If we define isomorphic groups as being "the same", there are two different groups of size 4. Give an example of each group as a Cayley table. Hint: Check all the conditions (axioms) for a group. There will be a deduction if you use the same elements of the group as another student.

[都合により削除]

Algebraic Structures Related to Groups

Abelian group: Group where the operation is also commutative
Examples:
- All groups of size 3 or 4
- (ℤ; +): Integers with addition
- (ℝ-{0}; ·) Reals (excluding 0) with multiplication
Semigroup: Group without unit element and without inverse elements

More Algebraic Structures

Ring: Two operations, 'addition' and 'multiplication'; forms an Abelian group under addition and a semigroup under multiplication, and addition distributes over multiplication.
Examples: ℝ, ring of polynomials
Field: Same as ring, but commutative, and 'multiplication' has to have an inverse for all non-zero elements.
Lattice: Two operations; both operations are associative and commutative, and respect absorption laws
Boolean Algebra

Boolean Algebra

A Boolean algebra is an algebraic structure
A Boolean algebra has one set (B), two special elements (0, 1), and three operations (⫬, △, ▽)
B = (B, 0, 1, ⫬, △, ▽) is a Boolean algebra if and only if it meets the following conditions:
- 0 ∈ B, 1 ∈ B (1 is called unit element, 0 is called zero element)
- ⫬ is an unary operation on B, △ and ▽ are binary operations on B
- △ (called join) and ▽(called meet) are associative and commutative, and distribute over each other
- ∀a∈B: a▽0 = a, a△1 = a (0 is the neutral element for ▽, 1 is the neutral element for △)
- ∀a∈B: a▽⫬a = 1, a△⫬a = 0
- Axioms (Huntington axioms (1904), see lecture 5):
  a▽0 = a [+D]; a▽b=b▽a [+D];
  a▽(b△c)=(a▽b)△(a▽c) [+D]; a▽⫬a=1 [+D]
This definition (operations, axioms) can be simplified
Boolean algebras define a (partial) order, with 1 at the top and 0 at the bottom (∀a∈B: 0≤a≤1)

Boolean Algebra Example 1:
Basic Logic

B is the set {false, true}
0 is false (0)
1 is true (1)
⫬ is negation (logical not, ¬)
△ is conjunction (logical and, ∧)
▽ is disjunction (logical or, ∨)
The (partial) order relation uses false<true , or a≤b ⇔ a→b
Hasse diagram: BAbasic_logic.svg

Comments on Boolean Algebra

Symbols 0, 1 (bold), ⫬, △, and ▽ are abstract and do not have any concrete meaning
A Boolean algebra is a (partial) order and a lattice
Boolean algebra is a general concept covering logical operations, set operations, some (partial) orders, ... (see examples)
Because a Boolean algebra is a partial order, it can be shown with a Hasse diagram

Boolean Algebra Example 2:
A Powerset with Set Operations

B is the powerset of a (finite) set U (B = P(U))
0 is the empty set ({})
1 is U (the universal set)
⫬ is the complementary set operation
△ is set intersection (∩)
▽ is set union (∪)
The (partial) order relation is ⊂ (subset)
Hasse diagram for U={a, b, c}: BApowerset.svg

Bitwise Operations

Inside a computer, information is represented as strings of bits (e.g. a=00101011, b=10011100).
Bitwise operations are bitwise logic operations.
Bitwise not: ¬ is applied to each bit (~ in C^*, e.g. ¬_bitwisea=11010100)
Bitwise and: ∧ is applied to bits in the same position (& in C^*, e.g. a∧_bitwiseb=00001000)
Bitwise or: ∨ is applied to bits in the same position (| in C^*, e.g. a∨_bitwiseb=10111111)
Bitwise xor: XOR (⊕) is applied to bits in the same position (^ in C^*, e.g. a⊕_bitwiseb=10110111)

^*) and many other programming languages

Boolean Algebra Example 3:
Bitstrings and Bitwise Operations

B is the set of all the bit strings of length n (e.g. for n=2, B={00, 01, 10, 11})
0 is the bit string of length n with all bits set to 0 (e.g. 00)
1 is the bit string of length n with all bits set to 1 (e.g. 11)
⫬ is bitwise not
△ is bitwise and
▽ is bitwise or
The (partial) order relation is the conjunction of the order relation at each bit position
(a≤b ↔ ∀i∈{1,...,n}: a_i≤b_i; a has 0 in all those bit positions where b has 0; in C: ~(b|~a) == 0).
Hasse diagram for bit strings of length 4: BAbitstrings.svg

Boolean Algebra Example 4:
Integers and Divisibility

B is formed by taking a set of n pairwise coprime integers^* (e.g. 2, 3, 5), and all the multiples that include or exclude each of the integers (including 1) (e.g. B = {1, 2, 3, 5, 6, 10, 15, 30})
0 is 1
1 is the product of all the original n integers (e.g. 30)
⫬a is 1/a
△ is the greatest common divisor (GCD) of the operands
▽ is the least common multiple (LCM) of the operands
The (partial) order relation is divisibility
Hasse diagram for {1, 2, 3, 5, 6, 10, 15, 30}: BAdivisibility.svg

^*) This restriction can be slightly relaxed.

The Structure of Boolean Algebras

The size of a finite boolean algebra is always |B| = 2ⁿ
All boolean algebras of the same size have the same structure (an n-dimensional cube)
(animation of a 4-dimensional cube, even higher dimensions (code))
The Hasse diagram is the edges of the n-dimensional cube, when placing 0 at the bottom and 1 at the top
The result of △ is the highest vertex equal or lower than both operands (infimum, greatest lower bound)
The result of ▽ is the lowest vertex equal or higher than both operands (supremum, least upper bound)
The result of ⫬ is the vertex diagonally oposite of its operand
All boolean algebras of dimension n are isomorphic (have the same structure)
For all boolean algebras, the same laws apply (they can be proven from the axioms)

Isomorphisms for Examples

Basic logic is isomorphic to bitstrings of length 1 (0=false, 1=true)
The subsets of a set of size n can be expressed as bit strings of size n:
Each bit indicates presence (1) or absence (0) of the respective element from the subset
Set operations can be defined by logical operations
Examples: A ∪ B = {x| x∈A ∨ x∈B}; A ∩ B = {x| x∈A ∧ x∈B}; A^c = {x| ¬(x∈A)}
Examples with divisibility are constructed to include/exclude each factor; this can be mapped to bitstrings or to set operations

Axioms for Boolean Algebras

The axioms for Boolean algebras are the same as the axioms for basic logic (standard/Huntington/Robbins/Sheffer/Wolfram).

There is a choice between compactness and obviousness.

We obtained the axioms by starting with basic logic and trying to find axiomatizations.

We obtain Boolean algebras by trying to find all objects that conform to these axioms.

Basic logic ⇒ Axioms of basic logic/boolean algebras ⇒ Boolean algebras

The Magic Garden of George B.

(The Magic Garden of George B. And Other Logic Puzzles, Raymond Smullyan, Polimetrica, 2007)

A very special garden: Flowers can change colors every day
(set of flowers F, set of days D, color of flower a on day d: color(a, d))
Each flower is either blue or red for a whole day
(∀f∈F: ∀d∈D: color(f,d)=red⊕color(f,d)=blue)
For any flowers a and b, there is a flower c that is red on all and only those days on which a and b are both blue.
(∀a, b∈F: ∃c∈F: ∀d∈D: color(a,d)=color(b,d)=blue ↔ color(c,d)=red)
Any two different flowers a and b have different colors on at least one day.
(∀a, b∈F: ∃d∈D: a≠b→color(a,d)≠color(b,d))
We know that the garden has more than 200 but less than 500 flowers. (200<|F|<500)
Question 1: How many flowers does the garden have?
Question 2: What is the minimum of days that the flowers bloom?
Question 3: Who is George B.?

How to Solve the Magic Garden Puzzle

Let red stand for 1, blue for 0
Let each day be a bit position
Let each flower be a bit combination
"there is a flower c that is red on all and only those days on which a and b are both blue" means that bitwise NOR is always defined
Because any (bitwise) logical operation can be defined based on bitwise NOR, all logical operations are defined
The flowers therefore form a Boolean algebra

The Magic Garden Boolean Algebra

B is F (set of flowers)
0 is a flower that is always^(*) blue
1 is a flower that is always^(*) red (*assuming |F|=2^|D|)
⫬a is the flower that is red on those days where a is blue and blue on those days where a is red
△ is the flower that is red on those days where both of its operands are red , otherwise blue
▽ is the flower that is red on those days where one or more of its operands are red , otherwise blue
The (partial) order relation is: a flower a is smaller or equal than a flower b if on all days where b is blue, a is also blue
Hasse diagram for 32 flowers: BAgarden.svg

Answers to Magic Garden Quiz

The size of a finite Boolean algebra is 2ⁿ, hence 200<|F|<500 → |F|=256=2⁸; min(|D|)=8
Question 1: How many flowers does the garden have? 256
Question 2: What is the minimum of days that the flowers bloom? 8 days (dimension of this Boolean algebra)
Question 3: Who is George B.? George Boole, after which Boolean algebra is named

Summary

Boolean algebra is (an example of) an algebraic structure
Boolean algebras have one set (B), two special elements (0 and 1), and three operations (⫬, △, and ▽)
The axioms of boolean algebra are the same as the axioms of basic logic; there are various choices with tradeoffs between obviousness and compactness
Boolean algebras are (partial) orders and lattices
There are many examples of Boolean algebras: basic logic (with logic operations), bit strings (with bitwise logic operations), powersets (with set operations), certain sets of integers (with divisibility), gardens with flowers,...
All finite Boolean algebras are of size 2ⁿ, and are isomorphic to an n-dimensional cube

Homework

Deadline: December 14, 2022 (Thursday), 18:40.

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)

Draw the Hasse diagram of a Boolean algebra of dimension 4 (16 elements). There will be a deduction if you use the same elements as another student.

Additional Homework (no need to submit): Which Boolean function makes a group out of any Boolean algebra?

Glossary

coverage: (試験) 範囲
cyclic group: 巡回群
Klein four-group: クラインの四元群
Abelian group: アベル群、可換群
semigroup: 半群
ring: 環 (かん)
polynomial: 多項式
field: 体 (たい)
lattice: 束 (そく)
Boolean algebra: ブール代数
zero element: 零元
unary operation: 単項演算
binary operation: 二項演算
bitwise operation: ビット毎演算
bitwise not: ビット毎否定
bitwise and: ビット毎論理積
bitwise or: ビット毎論理和
bitwise xor (exclusive or): ビット毎排他的又は
coprime: 互いに素
greatest common divisor: 最大公約数
least common multiple: 最小公倍数
coprime: 互いに素
pairwise coprime: 対ごとに素、どの二つも互いに素
n-dimensional: n 次元 (の)
cube: 立方体
supremum (least upper bound): 上限、最小上界
infimum (greatest lower bound): 下限、最大下界