Problem $5[20$ pts$]$

Consider the situation where the set of men is $\backslash \{$Cole$,$ Jack, Ken, Larry$\backslash \},$ and the set of women is $\backslash \{$Gail$,$ Heather, Jane, Maggie$\backslash \}.$ Preference lists for men and women are given in the two tables below.

We see that Cole's first choice is Heather, his second choice is Jane, his third is Maggie, and his four this Gail. The preferences of the others are read in the same way.

Suppose you have the following list of matching solutions:

$($i$)\backslash \{($Cole$,$ Heather$),($Jack$,$ Jane$),($Ken$,$ Gail$),($Larry$,$ Maggie$)\backslash \}$

$($ii$)\backslash \{($Cole$,$ Heather$),($Jack$,$ Gail$),($Ken$,$ Maggie$),($Larry$,$ Jane$)\backslash \}$

$($iii$)\backslash \{($Cole$,$ Heather$),($Jack$,$ Maggie$),($Ken$,$ Jane$),($Larry$,$ Gail$)\backslash \}$

$($iv$)\backslash \{($Cole$,$ Heather$),($Jack$,$ Jane$),($Ken$,$ Maggie$),($Larry$,$ Gail$)\backslash \}$

Which one is a stable marriage $($show the calculation details$)?$

$($a$)$ i and ii

$($b$)$ ii and iii

$($c$)$ ii and iv

$($d$)$ iii and iv

$($e$)$ i and iv