Now that you have practiced the basic algorithm, let’s study the stable marriage problem a little bit quantitavely. here we define the following notation: on day j, let pj(m) be the rank of the woman that man m proposes to (where the first woman on his list has rank 1 and the last has rank n). also, let rj(w) be the total number of men that woman w has rejected up through day j − 1 (i. e. not including the proposals on day j). answer the following questions using the notation above.(a) prove or disprove the following claim: ∑m pj(m)−∑w rj(w) is independent of j. if it is true, also give the value of ∑m pj(m)−∑w rj(w). the notation, ∑m and ∑w , simply means that we are summing over all men and all women.(b) prove or disprove the following claim: one of the men or women must be matched to someone who is ranked in the top half of their preference list. you may assume that n is even.