Boolean Algebra

(ブール代数)

Discrete Mathematics I

12th lecture, Dec. 19, 2014

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture12.html

Martin J. Dürst

AGU

© 2005-14 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。

 

Final Exam・期末試験

Coverage:
Complete contents of lecture and handouts
Past exams: 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013
How to view example solutions:
Use Opera 12.17 (Windows/Mac/Linux)
For Google Chrome, install Style Chooser
For Firefox, install Context Style Switcher
Select solutions style (e.g. View → Style → solutions)
Some solutions are missing
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

 

Last Week's Homework:
Symmetric Group of Order 3

Create a table of the symmetric group of order 3. Structure the table so that it looks like a multiplication table, with a row and a column for each permutation. Use lexical order for the permutations. (Use the row headings as the left operand, the column headings as the right operand.)

[都合により削除]

 

Group Isomorphism

G e a b
e e a b
a a b e
b b e a
K 0 2 1
0 0 2 1
2 2 1 0
1 1 0 2
H 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

 

Boolean Algebra

 

Comments on Boolean Algebra

 

Example of Boolean Algebra:
Basic Logic

 

Example of Boolean Algebra:
A Powerset with Set Operations

 

Bitwise Operations

 

Example of Boolean Algebra:
Bitstrings and Bitwise Operations

 

Example of Boolean Algebra:
Integers and Divisibility

 

The Structure of Boolean Algebras

 

Isomorphisms for Examples

 

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)

 

How to Solve the Magic Garden Puzzle

 

This Week's Homework

Deadline: January 8, 2015 (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)

Homework 1: 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.

Homework 2: 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.

Homework 3 (don't submit): Prepare for final exam using past exams.

 

Glossary

coverage
試験範囲
necessary condition
必要条件
sufficient condition
十分条件
group isomorphism
群同形
isomorphic
同形の、同型の
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
greatest common divisor
最大公約数
least common multiple
最小公倍数
n-dimensional
n 次元 (の)
cube
立方体