Consider the Gale-Shapley algorithm with equal number of men and women where men propose to women. We say that a man can manipulate the outcome by switching the ranking of some women in his preference list and end up with a partner that he prefers more than his partner under his true preference list.

a) Given a fixed preference list for all women and all other men, can a man improve his partner by changing his preference list and get matched up with a partner that he prefers to the one under his true preference list? For example, a man may change the ranking of two women w and w' , run the Gale-Shapley algorithm, and end up with a partner w'' that he prefers more to his initial partner?

b.) Focusing on women, is there a switch that would improve the partner of woman who switched her preferences, given a fixed preference list for all men and all other women?

For each parts of this question, you should either show an example of such manipulation or provide a proof of showing that manipulation is impossible.