operations on relations

operations on relations

Recall that an

n

-ary

relation

(

n

>

0

) is a subset of a

product

of some

n

sets. In this definition, any

n

-ary relation for which

n

>

1

is automatically an

(

n

-

1

)

-ary relation, and consequently a

binary relation

. On the other hand, a unary, or

1

-ary relation, being the subset

B

of some set

A

, can be viewed as a binary relation (either realized as

B

×

B

or

Δ

B

:=

{

(

b

,

b

)

∣

b

∈

B

}

) on

A

. Hence, we shall concentrate our discussion on the most important kind of relations, the binary relations, in this entry.

Compositions, Inverses, and Complements

Let

ρ

⊆

A

×

B

and

σ

⊆

B

×

C

be two binary relations. We define the

composition

of

ρ

and

σ

, the

inverse

(or

converse

) and the

complement

of

ρ

by

•

ρ

∘

σ

:=

{

(

a

,

c

)

∣

(

a

,

b

)

∈

ρ

⁢

and

⁢

(

b

,

c

)

∈

σ

⁢

for some

⁢

b

∈

B

}

,

•

ρ

-

1

:=

{

(

b

,

a

)

∣

(

a

,

b

)

∈

ρ

}

, and

•

ρ

′

:=

{

(

a

,

b

)

∣

(

a

,

b

)

∉

ρ

}

.

ρ

∘

σ

,

ρ

-

1

and

ρ

′

are binary relations of

A

×

C

,

B

×

A

and

A

×

B

respectively.

Properties

.
Suppose

ρ

,

σ

,

τ

are binary relations of

A

×

B

,

B

×

C

,

C

×

D

respectively.

1.

Associativity

of

relational compositions

:

(

ρ

∘

σ

)

∘

τ

=

ρ

∘

(

σ

∘

τ

)

.

2.

(

ρ

∘

σ

)

-

1

=

σ

-

1

∘

ρ

-

1

.

3.

For the rest of the remarks, we assume

A

=

B

. Define the

power

of

ρ

recursively by

ρ

1

=

ρ

, and

ρ

n

+

1

=

ρ

∘

ρ

n

. By 1 above,

ρ

n

+

m

=

ρ

n

∘

ρ

m

for

m

,

n

∈

ℕ

.

4.

ρ

-

n

may be also be defined, in terms of

ρ

-

1

for

n

>

0

.

5.

However,

ρ

n

+

m

≠

ρ

n

∘

ρ

m

for

all

integers, since

ρ

-

1

∘

ρ

≠

ρ

∘

ρ

-

1

in general.

6.

Nevertheless, we may define

Δ

:=

{

(

a

,

a

)

∣

a

∈

A

}

. This is called the

identity

or

diagonal relation

on

A

. It has the property that

Δ

∘

ρ

=

ρ

∘

Δ

=

ρ

. For this, we may define, for every relation

ρ

on

A

,

ρ

0

:=

Δ

.

7.

Let

ℛ

be the set of all binary relations on a set

A

. Then

(

ℛ

,

∘

)

is a monoid (

http://planetmath.org/Monoid

) with

Δ

as the

identity element

.

Let

ρ

be a binary relation on a set

A

, below are some special relations

definable

from

ρ

:

•

ρ

is

reflexive

if

Δ

⊆

ρ

;

•

ρ

is

symmetric

if

ρ

=

ρ

-

1

;

•

ρ

is anti-symmetric if

ρ

∩

ρ

-

1

⊆

Δ

;

•

ρ

is

transitive

if

ρ

∘

ρ

⊆

ρ

;

•

ρ

is a

pre-order

if it is reflexive and transitive

•

ρ

is a

partial order

if it is a pre-order and is anti-symmetric

•

ρ

is an equivalence if it is a pre-order and is symmetric

Of these, only

reflexivity

is preserved by

∘

and both

symmetry

and anti-symmetry are preserved by the inverse

operation

.

Other Operations

Some operations on binary relations yield unary relations. The most common ones are the following:

Given a binary relation

ρ

∈

A

×

B

, and an

elements

a

∈

A

and

b

∈

B

, define

ρ

⁢

(

a

,

⋅

)

:=

{

y

∣

(

a

,

y

)

∈

ρ

}

and

ρ

⁢

(

⋅

,

b

)

:=

{

x

∣

(

x

,

b

)

∈

ρ

}

.

ρ

⁢

(

a

,

⋅

)

⊆

B

is called the

section

of

ρ

in

B

based at

a

, and

ρ

⁢

(

⋅

,

b

)

⊆

A

is the

section of

ρ

in

A

(based) at

b

. When the

base points

are not mentioned, we simply say a section of

ρ

in

A

or

B

, or an

A

-section or a

B

-section of

ρ

.

Finally, we define the domain and range of a binary relation

ρ

⊆

A

×

B

, to be

•

dom

⁡

(

ρ

)

:=

{

a

∣

(

a

,

b

)

∈

ρ

⁢

for some

⁢

b

∈

B

}

, and

•

ran

⁡

(

ρ

)

:=

{

b

∣

(

a

,

b

)

∈

ρ

⁢

for some

⁢

a

∈

A

}

.

respectively. When

ρ

is a function, the domain and range of

ρ

coincide with the domain of range of

ρ

as a function.

Remarks

. Given a binary relation

ρ

⊆

A

×

B

.

1.

ρ

⁢

(

a

,

⋅

)

, realized as a binary relation

{

a

}

×

ρ

⁢

(

a

,

⋅

)

can be viewed as the composition of

Δ

a

∘

ρ

, where

Δ

a

=

{

(

a

,

a

)

}

. Similarly,

ρ

∘

Δ

b

=

ρ

⁢

(

⋅

,

b

)

×

{

b

}

.

2.

dom

⁡

(

ρ

)

is the

disjoint union

of

A

-sections of

ρ

and

ran

⁡

(

ρ

)

is the disjoint union of

B

-sections of

ρ

.

3.

dom

⁡

(

ρ

-

1

)

=

ran

⁡

(

ρ

)

and

ran

⁡

(

ρ

-

1

)

=

dom

⁡

(

ρ

)

.

4.

ρ

∘

ρ

-

1

=

dom

⁡

(

ρ

)

×

dom

⁡

(

ρ

)

and

ρ

-

1

∘

ρ

=

ran

⁡

(

ρ

)

×

ran

⁡

(

ρ

)

.

5.

If

A

=

B

, then

dom

⁡

(

ρ

)

×

A

=

ρ

∘

A

2

and

ran

⁡

(

ρ

)

=

A

2

∘

ρ

.

6.

The composition of binary relations can be generalized: let

R

be a subset of

A

1

×

⋯

×

A

n

and

S

be a subset of

B

1

×

⋯

×

B

m

, where

m

,

n

are positive integers. Further, we assume that

A

n

=

B

1

=

C

. Define

R

∘

S

, as a subset of

A

1

×

⋯

×

A

n

-

1

×

B

2

×

⋯

⁢

B

m

, to be the set

{

(

a

1

,

…

,

a

n

-

1

,

b

2

,

…

,

b

m

)

∣

∃

c

∈

C

⁢

with

⁢

(

a

1

,

…

,

a

n

-

1

,

c

)

∈

R

⁢

and

⁢

(

c

,

b

2

,

…

,

b

m

)

∈

S

}

.

If

m

=

n

=

2

, then we have the familiar composition of binary relations. If

m

=

1

and

n

=

2

, then

R

∘

S

=

{

a

∣

(

a

,

c

)

∈

R

⁢

and

⁢

c

∈

S

}

. The case where

m

=

2

and

n

=

1

is

similar

. If

m

=

n

=

1

, then

R

∘

S

=

{

true

}

if

R

∩

S

≠

∅

. Otherwise, it is set to

{

false

}

.

Title

operations on relations

Canonical name

OperationsOnRelations

Date of creation

2013-03-22 16:26:40

Last modified on

2013-03-22 16:26:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A02

Classification

msc 03E20

Synonym

inverse relation

Synonym

relation composition

Synonym

converse of a relation

Synonym

diagonal relation

Related topic

GeneralAssociativity

Related topic

RelationAlgebra

Defines

power of a relation

Defines

relational composition

Defines

inverse of a relation

Defines

section

Defines

domain

Defines

range

Defines

identity relation

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