# Properties of Relation Not Preserved by Restriction

## Theorem

If a relation is:

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

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

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

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

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

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 $T$.

$\blacksquare$