Boolean Algebras

(ブール代数)

Discrete Mathematics I

12th lecture, December 21, 2018

http://www.sw.it.aoyama.ac.jp/2018/Math1/lecture12.html

Martin J. Dürst

© 2005-18 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

@@@@ next year, check/add the concept of order (as opposed to size) of a Boolean algebra (needs more research, "order" doesn't show on Wikipedia, not even "size" shows, maybe 'dimension' might work)

• Remaining schedule
• Summary and homework for last lecture
• Algebraic structures related to groups
• Boolean algebra
• Examples of boolean algebras
• Bitstrings and Bitwise Operations
• Structure of Boolean algebras

Video

Please use the video recordings!

Email messages will be sent out when the video recordings are available.

Usually, videos become available on the afternoon of the lecture.

Remaining Schedule

• December 21: this lecture
• January 11: 13th lecture
• January 15 (Friday lectures on a Tuesday): 14th lecture
• January 18: 15th lecture (makeup class)
• January 25, 11:10-12:35: Term final exam

  Please use your winter vacation to prepare seriously for the term final exam!

About makeup classes: The material in the makeup class is part of the final exam. If you have another makeup class at the same time, please inform the teacher as soon as possible.

補講について: 補講の内容は期末試験の対象。補講が別の補講とぶつかる場合には事前に申し出ること。

(注: 一時期、佐久田先生の授業とダブっていたが、解消済み。)

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)

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

Sorry, it was removed! :)

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

Sorry, it was removed! :)

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 absorbtion laws

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
  • △ and ▽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
• This definition (operations, axioms) can be simplified
• Boolean algebras define a (partial) order, with 1 at the top and 0 at the bottom

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 is false<true
• Hasse diagram: BAbasic_logic.svg

Comments on Boolean Algebra

• Symbols such as 0 and 1 (bold) 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, and some (partial) orders
• 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
• 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. ¬a=11010100)
• Bitwise and: ∧ is applied to bits in the same position (& in C^*, e.g. a∧b=00001000)
• Bitwise or: ∨ is applied to bits in the same position (| in C^*, e.g. a∨b=10111111)
• Bitwise xor: XOR (⊕) is applied to bits in the same position (^ in C^*, e.g. a⊕b=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, or 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)
• 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 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 its operands (infimum, greatest lower bound)
• The result of ▽ is the lowest vertex equal or higher than its operands (supremum, least upper bound)
• The result of ⫬ is the vertex diagonally oposite of its operand
• All boolean algebras of dimension n are isomorphic
• 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.

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 a day be a bit position
• Let a 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 logical operation can be defined based on bitwise NOR, all logical operations are defined
• The flowers therefore form a 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 only on those days where both its operands are red, otherwise blue
  • ▽ is the flower that is red whenever 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
• 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

This Week's Homework

Deadline: January 10, 2017 (Thursday), 19:00.

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 of the group as another student.

Additional Homework (no need to submit): Prepare for final exam using past exams.

Glossary

coverage

試験範囲

group isomorphism

群同形

isomorphic

同形の、同型の

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