apartness relation

apartness relation

Given two arbitrary

real numbers

r

,

s

, how does one (better yet, a computer) prove

r

=

s

? Given decimal representations of

r

and

s

:

r

=

r

0

.

r

1

⁢

r

2

⁢

r

3

⁢

…

s

=

s

0

.

s

1

⁢

s

2

⁢

s

3

⁢

…

one needs to show that

r

0

=

s

0

,

r

1

=

s

1

,

r

2

=

s

2

, etc… which involves demonstrating an

infinite

number of

equalities

. On the other hand, showing that

r

≠

s

is a finitary process, in that one proceeds as above, and then stops when he/she finds a decimal place

n

where the two corresponding digits differ:

r

n

≠

s

n

. Therefore, in constructive mathematics,

inequality

is a more appropriate choice of study than equality. The formal notion of inequality in constructive mathematics is that of an

apartness relation

.

An

apartness relation

on a set

X

is a

binary relation

#

on

X

satisfying the following conditions:

1.

∀

x

⁢

(

¬

⁢

x

⁢

#

⁢

x

)

,

2.

∀

x

∀

y

(

x

#

y

→

y

#

x

)

, and

3.

∀

x

∀

y

∀

z

(

x

#

y

→

(

x

#

z

∨

y

#

z

)

)

.

And if in

addition

, we have the following

4.

∀

x

∀

y

(

¬

x

#

y

↔

x

=

y

)

Then

#

is said to be

tight

.

In classical mathematics where the law of the excluded middle is accepted, the following are true:

1.

the first condition is

redundant

if

#

is assumed to be tight.

2.

the second condition above is redundant, for if

x

⁢

#

⁢

y

, then

x

⁢

#

⁢

x

or

y

⁢

#

⁢

x

by condition 3, but then by condition 1,

x

⁢

#

⁢

x

is false, therefore

y

⁢

#

⁢

x

.

3.

the

converse

of an apartness relation is an

equivalence relation

: define

x

⁢

R

⁢

y

iff not

x

⁢

#

⁢

y

. Since

x

⁢

#

⁢

x

is false for all

x

∈

X

,

x

⁢

R

⁢

x

for all

x

∈

X

. Also, the

contrapositive

of the second condition above shows that

R

is

symmetric

. Now, suppose

x

⁢

R

⁢

y

and

y

⁢

R

⁢

z

. Then not

x

⁢

#

⁢

y

and not

y

⁢

#

⁢

z

, or not

(

x

⁢

#

⁢

y

∨

z

⁢

#

⁢

y

)

. Applying the contrapositive of the third condition, we have not

x

⁢

#

⁢

z

, or

x

⁢

R

⁢

z

. So

R

is

transitive

.

4.

Conversely, the converse of an equivalence relation is an apartness relation: suppose

R

is an equivalence relation, and define

x

⁢

#

⁢

y

iff not

x

⁢

R

⁢

y

. Then irreflexivity and symmetry of

#

are clear from the

reflexivity

and symmetry of

R

. Now, suppose

x

⁢

#

⁢

y

. Pick any

z

∈

X

. If not

x

⁢

#

⁢

z

and not

y

⁢

#

⁢

z

, then

x

⁢

R

⁢

z

and

y

⁢

R

⁢

z

, or

z

⁢

R

⁢

y

, which means

x

⁢

R

⁢

y

since

R

is transitive. But this contradicts the

assumption

x

⁢

#

⁢

y

, or not

x

⁢

R

⁢

y

.

Based on the last two observations, we see that the notion of apartness is not a very useful in mathematics based on

classical logic

, as the study of equivalence relation suffices.

On the other hand, arguing constructively, the second and the last statements above no longer hold, because the law of the excluded middle (and

proof by contradiction

) is used to derive both results. In other words, the apartness relation is a distinct

concept

in constructive mathematics. Below are some examples (and non-examples) of tight apartness relations:

•

≠

is a tight apartness relation on the set of

rationals

ℚ

.

•

more generally,

≠

is a tight apartness relation on a set

X

if “

=

” on

X

is a deciable

predicate

, that is, for any pair

x

,

y

∈

X

, either

x

=

y

or

x

≠

y

.

•

However,

≠

is not a tight apartness relation on the set of reals

ℝ

, since if

x

≠

y

, it may not be possible to prove constructively that, for any given real number

z

,

x

≠

z

or

y

≠

z

(because such a proof may be infinite in length).

•

However, if we define

x

⁢

#

⁢

y

iff there is a rational number

r

such that

(

x

<

r

<

y

)

∨

(

y

<

r

<

x

)

,

then

#

is a tight apartness relation. To see this, first note that

#

is

irreflexive

and symmetric. Also, it is not hard to see that the converse of

#

is

=

. Now, suppose

x

⁢

#

⁢

y

. Pick any real number

z

. If not

x

⁢

#

⁢

z

, then

x

=

z

, which means

y

⁢

#

⁢

z

. Similarly, if not

y

⁢

#

⁢

z

, then

x

⁢

#

⁢

z

.

Title

apartness relation

Canonical name

ApartnessRelation

Date of creation

2013-03-22 19:35:45

Last modified on

2013-03-22 19:35:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03F99

Classification

msc 03F65

Classification

msc 03F60

Defines

tight

\@unrecurse

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