Let and be sets. A binary relation from to is a subset of . If - - PowerPoint PPT Presentation

SLIDE 1 ✁ ✂✄ ☎ ✆✝ ✆ ✞ ✝ ☎ ✟ ✄ ✝ ✟ ☎ ✆ ✂ ✠ ✡ ☛ ☞✌ ✍ ✎ ✏ ✑ ✒ ✓ ✔ ✕ ✖✗ ✘ ✙ ✚ ✛ ✗ ✜ ✖ ✢ ✣ ✤ ✥ ✙ ✦✧ ★ ✙ ✦ ✩ ✪ ✫✬✭ ✬✮ ✯ ✰✱ ✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ❋

• ✆

❍ ■ ✝ ✁ ❏ ❑ ✂ ▲

Let

▼

and

◆

be sets. A binary relation from

▼

to

◆

is a subset

• f

▼ ❖ ◆

. If

P❘◗ ❙ ❚ ❯❘❱ ❲

, we write

◗ ❲ ❚

and say

◗

is related to

❚

by

❲

.

▲

A relation on the set

▼

is a relation from

▼

to

▼

.

▲

A relation

❲

• n a set

▼

is called reflexive if

P ◗ ❙ ◗ ❯❘❱ ❲

for every element

◗ ❱ ▼

.

▲

A relation

❲

• n a set

▼

is called symmetric if

P ❚ ❙ ◗ ❯❘❱ ❲

whenever

P ◗ ❙ ❚ ❯❘❱ ❲

, for

◗ ❙ ❚ ❱ ▼

.

▲

A relation

❲

• n a set

▼

such that

P❘◗ ❙ ❚ ❯❘❱ ❲

and

P ❚ ❙ ◗ ❯❘❱ ❲

• nly if

◗ ❳ ❚

, for

◗ ❙ ❚ ❱ ▼

, is called antisymmetric .

▲

A relation

❲

• n a set

▼

is called transitive if whenever

P ◗ ❙ ❚ ❯❘❱ ❲

and

P ❚ ❙ ❨ ❯❘❱ ❲

, then

P ◗ ❙ ❨ ❯❘❱ ❲

, for

◗ ❙ ❚ ❱ ▼

.

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ✾ ❩❬ ■ ❭ ❪ ❍ ✆ ✂ ▲ ❲ ❫ ❳ ❴ P ◗ ❙ ❚ ❯❵ ◗ ❛ ❚ ❜ ▲ ❲ ❝ ❳ ❴ P ◗ ❙ ❚ ❯❵ ◗ ❞ ❚ ❜ ▲ ❲ ❡ ❳ ❴ P ◗ ❙ ❚ ❯❵ ◗ ❳ ❚ ❢ ◗ ❳ ❣ ❚ ❜ ▲ ❲ ❤ ❳ ❴ P ◗ ❙ ❚ ❯❵ ◗ ❳ ❚ ❜ ▲ ❲ ✐ ❳ ❴ P ◗ ❙ ❚ ❯❵ ◗ ❳ ❚ ❥ ❦ ❜ ▲ ❲ ❧ ❳ ❴ P ◗ ❙ ❚ ❯❵ ◗ ❥ ❚ ❛ ♠ ❜ ✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ✽ ♥ ❏ ❭ ♦ ✁ ❑ ✁ ❑ ♣

• ✆

❍ ■ ✝ ✁ ❏ ❑ ✂ ▲

Let

❲

be a relation from a set

▼

to a set

◆

and

q

be a relation from

◆

to a set

r

. The composite of

❲

and

q

is the relation consisting of

• rdered pairs

P ◗ ❙ ❨ ❯

, where

◗ ❱ ▼ ❙ ❨ ❱ r

, and for which there exists an element

❚ ❱ ◆

such that

P ◗ ❙ ❚ ❯❘❱ ❲

and

P ❚ ❙ ❨ ❯ ❱ q

. We denote the composite of

❲

and

q

by

q s ❲

.

▲

Let

❲

be a relation on the set

▼

. The powers

❲✉t

,

✈ ❳ ❦ ❙ ✇ ❙ ♠ ❙ ① ① ①

, are defined inductively by

❲ ❫ ❳ ❲

and

❲②t ③ ❫ ❳ ❲②t s ❲

.

▲

Theorem : The relation

❲

• n a set

▼

is transitive if and only if

❲✉t ④ ❲

for

✈ ❳ ❦ ❙ ✇ ❙ ♠ ❙ ① ① ①

.

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ✼

SLIDE 2 ♥ ❍ ❏ ✂ ✟ ☎ ✆ ✂ ❏ ⑤

• ✆

❍ ■ ✝ ✁ ❏ ❑ ✂ ▲

Let

⑥

be a property of relations (transitivity, refexivity, symmetry). A relation

q

is losure of

❲

w.r.t.

⑥

if and only if

q

has property

⑥

,

q

contains

❲

, and

q

is a subset of every relation with property

⑥

containing

❲

.

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ⑦

• ✆

❍ ■ ✝ ✁ ❏ ❑ ✂ ■ ❑ ⑧ ⑨ ☎ ■ ❪❶⑩ ✂ ▲

A directed graph, or digraph, consists of a set

❷

• f vertices (or nodes)

together with a set

❸

• f ordered pairs of elements of

❷

called edges (or arcs).

▲

A path from

◗

to

❚

in the directed graph

❹

is a sequence of one or more edges

P❘❺ ❻ ❙ ❺ ❫ ❯ ❙ P ❺ ❫ ❙ ❺ ❝ ❯ ❙ ① ① ① P❘❺ t ❼ ❫ ❙ ❺ t ❯

in

❹

, where

❺ ❻ ❳ ◗

and

❺ t ❳ ❚

. This path is denoted by

❺ ❻ ❙ ❺ ❫ ❙ ① ① ① ❙ ❺ t

and has length

✈

. A path that begins and ends at the same vertex is called a circuit

• r cycle.

▲

There is a path from

◗

to

❚

in a relation

❲

is there is a sequence of elements

◗ ❙ ❺ ❫ ❙ ❺ ❝ ❙ ① ① ① ❺ t ❼ ❫ ❙ ❚

with

P ◗ ❙ ❺ ❫ ❯ ❽ ❲ ❙ P❘❺ ❫ ❙ ❺ ❝ ❯ ❽ ❲ ❙ ① ① ① ❙ P ❺ t ❼ ❫ ❙ ❚ ❯ ❽ ❲

.

▲

Theorem: Let

❲

be a relation on a set

▼

. There is a path of length

✈

from

◗

to

❚

if and only if

P ◗ ❙ ❚ ❯ ❽ ❲✉t

.

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ❊ ♥ ❏ ❑ ❑ ✆ ✄ ✝ ✁ ❾ ✁ ✝ ❿ ▲

Let

❲

be a relation on a set

▼

. The connectivity relation

❲➁➀

consists of pairs

P❘◗ ❙ ❚ ❯

such that there is a path between

◗

and

❚

in

❲

.

▲

Theorem: The transitive closure of a relation

❲

equals the connectivity relation

❲➁➀

.

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ❅ ➂ ■ ☎ ✝ ✁ ✝ ✁ ❏ ❑ ✂ ▲

We want to use relations to form partitions of a group of students. Each member of a subgroup is related to all other members of the subgroup, but to none of the members of the other subgroups.

▲

Use the following relations: Partition by the relation ”older than” Partition by the relation ”partners on some project with” Partition by the relation ”comes from same hometown as”

▲

Which of the groups will succeed in forming a partition? Why?

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ➃

SLIDE 3 ❩ ➄ ✟ ✁ ❾ ■ ❍ ✆ ❑ ✄ ✆

• ✆

❍ ■ ✝ ✁ ❏ ❑ ✂ ▲

A relation on a set

▼

is called an equivalence relation if it is reflexive, symmetric, and transitive. Two elements that are related by an equivalence relation are called equivalent.

▲

Let

❲

be an equivalence relation on a set

▼

. The set of all elements that are related to an element

◗

• f

▼

is called the equivalence class of

◗

.

➅ ◗ ➆ ➇

: equivalence class of

◗

w.r.t.

❲

. If

❚ ❽ ➅ ◗ ➆ ➇

then

❚

is representative of this equivalence class.

▲

Theorem: Let

❲

be an equivalence relation on a set

▼

. The following statements are equivalent: (1)

◗ ❲ ❚

(2)

➅ ◗ ➆ ❳ ➅ ❚ ➆

(3)

➅ ◗ ➆ ➈ ➅ ❚ ➆ ➉ ❳ ➊ ✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ➋ ❩ ➄ ✟ ✁ ❾ ■ ❍ ✆ ❑ ✄ ✆

• ✆

❍ ■ ✝ ✁ ❏ ❑ ✂ ■ ❑ ⑧ ➂ ■ ☎ ✝ ✁ ✝ ✁ ❏ ❑ ✂ ▲

A partition of a set

q

is a collection of disjoint nonempty subsets

▼ ➌ ❙ ➍ ❽ ➎

(where

➎

is an index set) of

q

that have

q

as their union:

▼ ➌ ➉ ❳ ➊

for

➍ ❽ ➎ ▼ ➌ ➈ ▼ ➏ ❳ ➊

, when

➍ ➉ ❳ ➐ ➑ ➌ ➒ ➓ ▼ ➌ ❳ q ▲

Theorem: Let

❲

be an equivalence relation on a set

q

. Then the equivalence classes of

❲

form a partition of

q

. Conversely, given a partition

❴ ▼ ➌ ❵ ➍ ❽ ➎ ❜

• f the set

q

, there is an equivalence relation

❲

that has the sets

▼ ➌ ❙ ➍ ❽ ➎

, as its equivalence classes.

✲✳✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ➔