A finite binary relation over a set S can be represented in ML by a list of

$2$

$-$

tuples. Write an

ML function transitive that checks whether a relation is transitive.

You may define auxiliary functions.

For instance,

$-$

$$transitive;

val it

$=$

$$fn :

$($

a

$*$

$$a

$)$

$$list

$-$

$>$

$$bool

$-$

$$transitive nil;

val it

$=$

$$true : bool

$-$

$$transitive

$[$

$($

$1$

$,$

$1$

$)$

$]$

;

val it

$=$

$$true : bool

$-$

$$transitive

$[$

$($

$1$

$,$

$2$

$)$

$,$

$($

$1$

$,$

$4$

$)$

$,$

$($

$1$

$,$

$3$

$)$

$]$

;

val it

$=$

$$true : bool

$-$

$$transitive

$[$

$($

$1$

$,$

$2$

$)$

$,$

$($

$1$

$,$

$4$

$)$

$,$

$($

$2$

$,$

$3$

$)$

$]$

;

val it

$=$

$$false : bool

$-$

$$transitive

$[$

$($

$1$

$,$

$2$

$)$

$,$

$($

$1$

$,$

$4$

$)$

$,$

$($

$2$

$,$

$3$

$)$

$,$

$($

$1$

$,$

$3$

$)$

$]$

;

val it

$=$

$$true : bool