(ブール代数)

https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture12.html

© 2005-19 Martin J. Dürst Aoyama Gakuin University

- 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

- December 20: this lecture
- January 10: 13th lecture
- January 17: 14th lecture (makeup class)
- January 24: 15th lecture
- January 31,
**11:10**-12:35: Term final examPlease 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。

Proofs for two group theorems: Uniqueness of inverse, cancellation law

`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)

Symmetric Group of Order 3

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

(1, 2, 3) | (1, 3, 2) | (2, 1, 3) | (2, 3, 1) | (3, 1, 2) | (3, 2, 1) | |
---|---|---|---|---|---|---|

(1, 2, 3) | (1, 2, 3) |
(1, 3, 2) | (2, 1, 3) | (2, 3, 1) | (3, 1, 2) | (3, 2, 1) |

(1, 3, 2) | (1, 3, 2) | (1, 2, 3) |
(3, 1, 2) | (3, 2, 1) | (2, 1, 3) | (2, 3, 1) |

(2, 1, 3) | (2, 1, 3) | (2, 3, 1) | (1, 2, 3) |
(1, 3, 2) | (3, 2, 1) | (3, 1, 2) |

(2, 3, 1) | (2, 3, 1) | (2, 1, 3) | (3, 2, 1) | (3, 1, 2) | (1, 2, 3) |
(1, 3, 2) |

(3, 1, 2) | (3, 1, 2) | (3, 2, 1) | (1, 3, 2) | (1, 2, 3) |
(2, 3, 1) | (2, 1, 3) |

(3, 2, 1) | (3, 2, 1) | (3, 1, 2) | (2, 3, 1) | (2, 1, 3) | (1, 3, 2) | (1, 2, 3) |

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.

Solution 1 (cyclic group of order 4)

Interesting examples: Addition modulo 4 (below), rotation by multiples of 90°, {1, 2, 3, 4} and multiplication modulo 5 (see next lecture)

0 | 1 | 2 | 3 | |
---|---|---|---|---|

0 | 0 | 1 | 2 | 3 |

1 | 1 | 2 | 3 | 0 |

2 | 2 | 3 | 0 | 1 |

3 | 3 | 0 | 1 | 2 |

Solution 2 (Klein group):

Interesting examples: bitstrings of length 2 with XOR (below), reflections and rotations by 180°

00 | 01 | 10 | 11 | |
---|---|---|---|---|

00 | 00 | 01 | 10 | 11 |

01 | 01 | 00 | 11 | 10 |

10 | 10 | 11 | 00 | 01 |

11 | 11 | 10 | 01 | 00 |

- 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

- 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

- 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

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

- 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, some (partial) orders, ...
- Because a Boolean algebra is a partial order, it can be shown with a Hasse diagram

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

- 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. ¬_{bitwise}`a`=11010100) - Bitwise and: ∧ is applied to bits in the same position
(
`&`

in C^{*}, e.g.`a`∧_{bitwise}`b`=00001000) - Bitwise or: ∨ is applied to bits in the same position
(
`|`

in C^{*}, e.g.`a`∨_{bitwise}`b`=10111111) - Bitwise xor: XOR (⊕) is applied to bits in the same position
(
`^`

in C^{*}, e.g.`a`⊕_{bitwise}`b`=10110111)

^{*)} and many other programming languages

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

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 size of a finite boolean algebra is always |
`B`| = 2^{n} - 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)

- 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

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. 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.?

- 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 (bitwise) 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 2
, hence 200<|^{n}`F`|<500 → |`F`|=256=2^{8}; 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

- Boolean algebra is a an example of 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 half-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 2
^{n}, and are isomorphic to an`n`-dimensional cube

Deadline: **January 6**, 2019 (Monday after New Year
vacations), 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)

By using formula manipulation, show that the Wolfram axiom of Boolean logic
(((` A`⊼

Hints: Simplify the left side to obtain the right side. There should be between 15 and 20 steps.

Deadline: January 9, 2019 (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 as another student.

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

- 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