Relations

(関係)

Discrete Mathematics I

9th lecture, Nov. 28, 2014

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture9.html

Martin J. Dürst

AGU

© 2005-14 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

 

Summary of Last Lecture

 

Homework from Last Lecture (1, 2)

1. Create a set with four elements. If you use the same elements as other students, a deduction of points will be applied.

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2. Create the powerset of the set you created in problem 1.

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Homework from Last Lecture (3)

3. For sets A of size zero to six, create a table of the sizes of the powersets (|P(A)|).

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Homework from Last Lecture (4, 5)

4. Express the relationship between the size of a set A and the size of its powerset P(A) as a formula.

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5. Explain the reason behind the formula in problem 4.

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Homework from Last Lecture (6)

6. Create a table that shows, for sets A of size zero to five, and for each n (size of sets in P(A)), the number of such sets.

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Pascal's Triangle

(Pascal's triangle)

Start with a single 1 in the first row, surrounded by zeroes
((0 ... 0) 1 (0 ... 0)). Create row by row by adding
the number above and to the left and the number above and to the right.

                  1
                1   1
              1   2   1
            1   3   3   1
          1   4   6   4   1
        1   5  10  10   5   1
      1   6  15  20  15   6   1
    1   7  21  35  35  21   7   1
  1   8  28  56  70  56  28   8   1

 

Subsets and Pascal's Triangle

 

Subsets and Combinations

 

Direct Formula for Combinations

(prove it as a homework)

 

Factorial

Notation: n!

Definition: n! = 1 · 2 · ... (n-1) · n = ∏ni=1 i

(∏ is called product)

Question:

1! = 1

0! = 1

 

Neutral Element of an Operation

(also unit element, identity element)

 

Relations

 

Importance of Relations for IT

 

Tuples

 

Cartesian Product

 

Definition of Relation

 

Representation of Relations

 

Matrix Representation

A relation between sets A and B is represented as a matrix where:

Matrix representation is suited for binary relations.

A matrix with only 1 or 0 as entries is called a logical matrix (also binary matrix, relation matrix, or Boolean matrix)

 

Table Representation

A relation between several sets is represented in a table as follows:

Table representation is suited for relations of any arity.

Table representation is suited for sparse relations
(relations with very few entries).

Table representation is used in relational databases.

 

Graph Representation

A relation between sets A and B is represented as a graph as follows:

Graph representation is suited for binary relations.

 

Inverse Relation

 

This Week's Homework

Deadline: December 6, 2014 (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)

  1. Prove nCm = n!/(m! (n-m)!) for n>0, 0<m<n-0 using nCm = n-1Cm-1 + n-1Cm
  2. Give three examples of relations from the real world that can be expressed as mathematical relations

 

Glossary

Pascal's triangle
パスカルの三角形
combinatorics
組合せ論
combination
組合せ
permutation
順列
repeated combination
重複組合せ
repeated permutation
重複順列
factorial
階乗
product (∏)
総乗、総積
neutral element
単位元
relational database
関係データベース
tuple
タプル
ordered pair
順序対
n-tuple
n 項組、n 字組
triple
三項組、三字組
quadruple
四項組、四字組
quintuple
五項組、五字組
sextuple
六項組、六字組
septuple
七項組、七字組
octuple
八項組、八字組
nonuple
九項組、九字組
Cartesian product (set)
直積 (集合)
definition
定義
divisible
割り切りが可能
binary relation
2項関係
ternary relation
3項関係
(binary) relation on A
A の中の関係、A の上の関係、A における関係
representation
表現
matrix
行列
row
column
列、欄
correspond to
と対応する
arity
アリティ
sparse
スパース、まばら (な)
vertex (plural: vertices)
頂点、節
edge
directed
有向 (の)
reflexive relation
反射的関係
(main) diagonal
(主) 対角線
symmetric relation
対称的関係
(matrix) transposition
(行列) 転置
sibling
兄弟 (姉妹も含む)
antisymmetric relation
反対称的関係
opposite
反対
asymmetric relation
非対称的関係
transitive relation
推移的関係
descendant
子孫
anchestor
先祖
inverse relation
逆関係