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Cоnsider а set оf n hоspitаls, а set of m students students, a ranked list of students for each hospital, and a ranked list of hospitals for each student. In class, we studied the problem of finding a stable matching of hospitals and students when n = m. Now consider the variant of this problem where m > n, i.e., there are more students than hospitals. We are interested in sets M of hospital-student pairs in which each of the n hospitals is assigned exactly one student and no two hospitals are assigned the same students. We call M a hospital-matching. A pair h-s is unstable with respect to M if either s is not assigned to any hospital in M and h prefers s over the student assigned to h in M or s is assigned to some hospital in M and h and s prefer each other over their assigned partners in M. M is stable if it has no unstable pair of Type (1) or Type (2). a. Prove that the set of pairs output by the Gale-Shapely algorithm when hospitals propose is a hospital-matching, i.e., the set satisfied conditions (I) and (II). Hint: think about how the algorithm runs.