[Q6: 24 points]. This question has 8 parts. Each part is worth 3 points. It presents a version of the so called Arrow '3 Impossibility Theorem in honor of Kenneth Arrow. The idea is to teach you one way to think about this theorem by asking you to answer some simple questions. In general, this theorem highlights the difculty of coming up with \"reasonable\" ways to aggre gate the views of multiple individuals to come up with a \"social choice\". For example, consider elections. \"Voters\" report their individual preferences about \"candidates\[5.1] Let R denote the set of all possible ways in which the three candidates can be strictly ranked. Write down the set R. What is its cardinality? Tomorrow, the Dean will receive two rankings, one from each of the professors. We can think of a possible situation that may arise tomorrow as a pair of possible rankings by the two professors. As of today, the set of all possible situations that may arise tomorrow is therefore X = R X R. An :1: E X can therefore be represented as (r1, m), where n denotes the ranking by 191 and r2 denotes the ranking by m. [5.2] What is the cardinality of set X? [5.3] Provide one example of an a: = (qr-hm) and an w' = (T'l') such that 02 is at the top of r1, r2, 7"], and 7'5 but It 7E 92'. [5.4] How many elements of set X are such that the two professors have the same candidate at the top of their rankings. Suppose the Dean wants to gure out today hOW to deal with any possible situation she may face tomorrow. This is crucial. It means the Dean wants to nd a function f that takes each possible situation in X as input and produces the name of the candidate in 0 who should be hired as the output. [5.5] What is the cardinality of the set .7: (X, Y) that contains all possible functions fromX=R>