Boolean Algebras

(ブール代数)

Discrete Mathematics I

12th lecture, December 20, 2018

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

Martin J. Dürst

AGU

© 2005-19 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

 

Remaining Schedule

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。

 

Leftovers of Last Lecture

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

 

Summary of Last Lecture

 

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.

  (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)

 

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.

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

 

Algebraic Structures Related to Groups

 

More Algebraic Structures

 

Boolean Algebra

 

Boolean Algebra Example 1:
Basic Logic

 

Comments on Boolean Algebra

 

Boolean Algebra Example 2:
A Powerset with Set Operations

 

Bitwise Operations

*) and many other programming languages

 

Boolean Algebra Example 3:
Bitstrings and Bitwise Operations

 

Boolean Algebra Example 4:
Integers and Divisibility

*) This restriction can be slightly relaxed.

 

The Structure of Boolean Algebras

 

Isomorphisms for Examples

 

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)

 

How to Solve the Magic Garden Puzzle

 

Summary

 

Homework Due January 6

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 (((AB)⊼C)⊼(A⊼((AC)⊼A))=C) is a tautology. For each simplification step, indicate which law(s) you used.

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

  

Homework Due January 9

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.

 

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