relation

relation

Binary Relations

Before describing what a

relation

is generally, let us define a more specific kind of a relation: a

binary relation

. Basically, a binary relation

R

involves objects coming from two

collections

A

,

B

, where the objects are paired up so that each pair consists of an object from

A

, and an object from

B

.

More formally, a

binary relation

is a subset

R

of the

Cartesian product

(

http://planetmath.org/CartesianProduct

) of two sets

A

and

B

. One may write

a

⁢

R

⁢

b

to indicate that the

ordered pair

(

a

,

b

)

is an element of

R

. A subset of

A

×

A

is simply called a binary relation on

A

. If

R

is a binary relation on

A

, then we write

a

1

⁢

R

⁢

a

2

⁢

R

⁢

a

3

⁢

⋯

⁢

a

n

-

1

⁢

R

⁢

a

n

to mean

a

1

⁢

R

⁢

a

2

,

a

2

⁢

R

⁢

a

3

,

…

,

and

a

n

-

1

⁢

R

⁢

a

n

.

Given a binary relation

R

⊆

A

×

B

, the

domain

dom

⁡

(

R

)

of

R

is the set of

elements

in

A

forming parts of the pairs in

R

. In other words,

dom

⁡

(

R

)

:=

{

x

∈

A

∣

(

x

,

y

)

∈

R

⁢

for some

⁢

y

∈

B

}

and the

range

ran

⁡

(

R

)

of

R

is the set of parts of pairs of

R

coming from

B

:

ran

⁡

(

R

)

:=

{

y

∈

B

∣

(

x

,

y

)

∈

R

⁢

for some

⁢

x

∈

A

}

.

An example of a binary relation is the less-than relation on the integers, i.e.,

<

⊆

ℤ

×

ℤ

.

(

1

,

2

)

∈

<

, but

(

2

,

1

)

∉

<

.

Remarks

.

1.

In

set theory

, a binary relation is simply a set of ordered pairs (of sets or classes, depending on the

axiom system

used). Notice that, unlike the previous definition, sets (or classes)

A

and

B

are not specified in advance. Given a (binary) relation

R

, the domain of

R

is the set (or class) of elements

a

such that

a

⁢

R

⁢

b

for some

b

, and the range of

R

is the set (or class) or elements

b

such that

a

⁢

R

⁢

b

for some

a

. The union and the domain and the range of

R

is called the

field

of

R

.

2.

It may be possible to define a relation over a class. For example, if

𝒞

is the class of all sets, then

∈

can be thought of as a binary relation on

𝒞

.

3.

In term rewriting theory, a binary relation on a set is sometimes called a

reduction

, and is written

→

. This is to signify that

a

→

b

means that the element

a

is being “

reduced

” to

b

via

→

.

Arbitrary Relations

From the definition of a binary relation, we can easily generalize it to that of an arbitrary relation. Since a binary relation involves two sets, an arbitrary relation involves an arbitrary collection of sets. More specifically, a

relation

R

is a subset of some Cartesian product (

http://planetmath.org/GeneralizedCartesianProduct

) of a collection of sets. In symbol, this is

R

⊆

∏

i

∈

I

A

i

where each

A

i

is a set,

indexed by

some set

I

.

From this more general definition, we see that a binary relation is just a relation where

I

has two elements. In

addition

, an

n

-ary relation

is a relation where the

cardinality

of

I

is

n

(

n

finite). In symbol, we have

R

⊆

∏

i

=

1

n

A

i

.

It is not hard to see that any

n

-ary relation where

n

>

1

can be viewed as a binary relation in

n

-

1

different ways, for

R

⊆

A

1

×

A

2

×

⋯

×

A

n

=

∏

i

=

1

j

A

i

×

∏

i

=

j

+

1

n

A

i

,

where

j

ranges from

1

through

n

-

1

.

A common name for a

3

-ary relation is a

ternary relation

. It is also possible to have a

1

-ary relation

, or commonly known as a

unary relation

, which is nothing but a subset of some set.

Remarks

.

1.

Following from the first remark from the previous

section

, relations of higher arity can be inductively defined: for

n

>

1

, an

(

n

+

1

)

-ary relation is a binary relation whose domain is an

n

-ary relation. In this setting, a “unary relation” and relations whose arity is of “arbitrary” cardinality are not defined.

2.

A relation can also be viewed as a

function

(which itself is a relation). Let

R

⊆

A

:=

∏

i

∈

I

A

i

. As a subset of

A

,

R

can be identified with the

characteristic function

χ

R

:

A

→

{

0

,

1

}

,

where

χ

R

⁢

(

x

)

=

1

iff

x

∈

R

and

χ

R

⁢

(

x

)

=

0

otherwise. Therefore, an

n

-ary relation is

equivalent

to an

(

n

+

1

)

-ary characteristic function. From this, one may say that a

0

-ary, or a

nullary relation

is a unary characteristic function. In other words, a nullary relation is just a an element in

{

0

,

1

}

(or truth/falsity).

Title

relation

Canonical name

Relation

Date of creation

2013-03-22 11:43:28

Last modified on

2013-03-22 11:43:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

33

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A02

Classification

msc 03E20

Classification

msc 82C35

Related topic

Poset

Related topic

PartialOrder

Related topic

TotalOrder

Related topic

OrderingRelation

Related topic

Function

Related topic

WellFoundedRelation

Related topic

Property2

Related topic

GroundedRelation

Related topic

RelationBetweenObjects

Defines

unary relation

Defines

binary relation

Defines

ternary relation

Defines

n

-ary relation

Defines

domain

Defines

range

Defines

nullary relation

Defines

field

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