Boolean Algebra

(ブール代数)

Discrete Mathematics I

12th lecture, Dec. 16, 2016

http://www.sw.it.aoyama.ac.jp/2016/Math1/lecture12.html

Martin J. Dürst

© 2005-16 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

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

Remaining Schedule

• December 16: this lecture
• January 6: 13th lecture
• January 13: 14th lecture (makeup class)
• January 20: 15th lecture
• January 26 to February 2: Final exams

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。

Final Exam・期末試験

Coverage:

Complete contents of lecture and handouts

Past exams: 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014

How to view example solutions:

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Important points:

Read problems carefully (distinguish between calculation, proof, explanation,...)

Be able to explain concepts in your own words
Be able to do calculations (base conversions, truth tables,...) speedily

Combine and apply knowledge from different lectures

Write clearly

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, and an identity element and inverse elements
• Examples of groups are (ℤ, +), (ℝ-{0}, ·), (ℝ⁺, ·), and symmetric groups of order n (permutations of n elements with composition)
• Finite groups are represented by Cayley tables, and compared using the concept of isomorphism

Correction: How to Check Transitive Relations?

• Definition: If and only if for all x, y, and z, xRy ∧ yRz ⇒ xRz, then R is transitive.
  (∀x, y, z: xRy ∧ yRz ⇒ xRz) ⇔ R is transitive
• Examples: =, ≤, ≥, <, >, descendant, anchestor, divisible, ...
• How to check: Compose R with itself. If and only if the result is a subset of R (R∘R ⊂ R), then R is transitive.
• Additional observations:
  • A relation for which R∘R = R is called idempotent
  • All idempotent relations are transitive
  • Relations that are reflexive and transitive are idempotent
  • There are some idempotent relations that are not reflexive

Correction: Cayley Tables

The identity element only appears once on the (main) diagonal, and is distributed symmetrically to the diagonal (reason: inverse element)

Algebraic Structures Related to Groups

• An Abelian group is a 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
• A semigroup is a 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
• 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
  • Definition (operations, axioms) can be simplified

Comments on Boolean Algebra

• Symbols such as 0 and 1 (bold) are abstract and don't 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

Example of Boolean Algebra:
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

Example of Boolean Algebra:
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.
• Bitwise operations are bitwise logic operations.
• Bitwise not: ¬ is applied to each bit (~ in C)
• Bitwise and: ∧ is applied to bits in the same position (& in C)
• Bitwise or: ∨ is applied to bits in the same position (| in C)
• Bitwise xor: XOR is applied to bits in the same position (^ in C)

Example of Boolean Algebra:
Bitstrings and Bitwise Operations

• B is the set of all the bit strings of length n
• 0 is the bit string of length n with all bits set to 0
• 1 is the bit string of length n with all bits set to 1
• ⫬ 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 ↔ a has 0 in all those bit positions where b has 0.
• Hasse diagram for bit strings of length 4: BAbitstrings.svg

Example of Boolean Algebra:
Integers and Divisibility

• B is formed by taking a set of n pairwise coprime integers and all the multiples that include or exclude each of the integers
• 0 is 1
• 1 is the product of all the original n integers
• ⫬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

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)
• 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 Algebra

The axioms for Boolean algebra 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 Algebra 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 (order of this Boolean algebra)
• Question 3: Who is George B.? George Boole, after which Boolean algebra is named

This Week's Homework

Deadline: January 6, 2017 (Friday), 10:30.

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 order 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

polynomial

多項式

field

体

lattice

束

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