Properties of Relation Not Preserved by Restriction

Theorem

If a relation is:

  • serial,
  • non-reflexive,
  • non-symmetric,
  • non-transitive or
  • non-connected

it is impossible to state without further information whether or not any restriction of that relation has the same properties.


Proof

Restriction of Serial Relation is Not Necessarily Serial

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, b}, \tuple {b, b} }$.

$\RR$ is a serial relation, as can be seen by definition.


Now let $T = \set a$.

Then:

$\RR {\restriction_T} = \O$

So:

$\not \exists y \in T: \tuple {a, y} \in \RR {\restriction_T}$

That is, $\RR {\restriction_T}$ is not a serial relation on $T$.

$\blacksquare$


Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {b, b} }$.

$\RR$ is a non-reflexive relation, as can be seen by definition:

$\tuple {a, a} \notin \RR$
$\tuple {b, b} \in \RR$

Now let $T = \set a$.

Then $\RR {\restriction_T} = \O$.

So:

$\forall x \in T: \tuple {x, x} \notin \RR {\restriction_T}$

That is, $\RR {\restriction_T}$ is an antireflexive relation on $T$.

That is, specifically not a non-reflexive relation.

$\blacksquare$


Restriction of Non-Symmetric Relation is Not Necessarily Non-Symmetric

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, b}, \tuple {b, b} }$.

$\RR$ is a non-symmetric relation, as can be seen by definition.

Now let $T = \set b$.

Then $\RR {\restriction_T} \ = \set {\tuple {b, b} }$.

So:

$\forall x, y \in T: \tuple {x, y} \in \RR {\restriction_T} \implies \tuple {y, x} \in \RR {\restriction_T}$

as can be seen by setting $x = y = b$.

So $\RR {\restriction_T}$ is a symmetric relation on $T$.

That is, $\RR {\restriction_T}$ is not a non-symmetric relation on $T$.

$\blacksquare$


Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, b}, \tuple {b, a}, \tuple {b, b} }$.

$\RR$ is a non-transitive relation, as can be seen by definition.

Now let $T = \set b$.

Then:

$\RR {\restriction_T} = \set {\tuple {b, b} }$

So:

$\forall x, y \in T: \tuple {x, y} \in \RR {\restriction_T} \land \tuple {y, z} \in \RR {\restriction_T} \implies \tuple {y, z} \in \RR {\restriction_T}$

as can be seen by setting $x = y = z = b$.

So $\RR {\restriction_T}$ is a transitive relation on $T$.

That is, $\RR {\restriction_T}$ is not a non-transitive relation on $T$.

$\blacksquare$


Restriction of Non-Connected Relation is Not Necessarily Non-Connected

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, a}, \tuple {b, b} }$.

$\RR$ is a non-connected relation, as can be seen by definition: neither $a \mathrel \RR b$ nor $b \mathrel \RR a$.

Now let $T = \set a$.

Then $\RR {\restriction_T} = \set {\tuple {a, a} }$.

Then $\RR {\restriction_T}$ is trivially connected on $T$.

$\blacksquare$


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