B.1) Valid Arguments . ..... . . . . / 4 points pVr Prove, by constructing the truth table, that the following argument form is valid: :. pv q Please indicate which columns are premises and which column is the conclusion. Also mark the critical row(s). B.2) Logic Equivalence . / 4 points Let p and q be two propositions. Use the laws of logic equivalence discussed in class (and in the textbook) to prove that (p = q) = p = p A (q = p). (Hint: p= q= -p V q)The following questions are on discrete probability and counting. Consider an experiment of selecting a 3-digit number uniformly at random from the set of all possible 3-digit numbers 000 - 999. The 1000 numbers are equally likely to be picked. B.3) Probability Space / 1.5 points Define the probability space (the sample space and the probability function) for the above experiment. B.4) Probability of Events 1.5 points What is the probability that the selected number is even? B.5) Probability of Events / 1.5 points Let A be the event that the first and the last digits are both greater than or equal to 8. Compute P(A).B.6) Conditional Probability / 1.5 points Let B be the event that a randomly selected 3-digit number is less than or equal to 809. Compute P(B| A), where A is the event defined in the previous question.Part C. Modelling and Logic Arguments (12 Points) Questions in this part all about the logic description of a round-robin tournament. Let I be the set of teams and assume that there is no tie games. A team a dominates another team v if and only if u beats v or u beats some other team that beats v. When answering the questions, you can use the following predicates, where the domain of the predicate variables is L. . K(u) = "u is a king", B(u, v) = "u beats v". . I(u, v) = "u beats some other team that beats team v", and . D(u, v) = "u dominates v", Write each of the following four statements in symbolic form. C.1) For every pair of teams u and v, either u beats v or v beats u, but not both. /1.5 points C.2) If a team o is a king, then it dominates all other teams. 1.5 points C.3) If a team u dominates a team v, but v beats u, then it must be the case that u beats some other team that beats v. . . . . . . 1.5 points C.4) The negation of I(u, v): "a beats some other team that beats team v" / 1.5 pointsC.5) Logic Arguments ... 6 points Let Canucks and Rockets be two particular teams and assume that the following four statements 51) - 84) are all true. (an)( + ()y '73 a pen A (IS $3) Vu and v E L, D(u, v) = B(u, v) V I(u, v) 52) K (Canucks) 54) -I(Canucks, Rockets) Did Canucks beat Rockets in the tournament? Construct a sound logic argument to justify your answer. You can use any inference rule in propositional logic and in predicate calculus discussed in class, including Universal Instantiation, (Universal) Modus Ponens/Tolens, (Universal) Transitivity, and Elimination. Attention - No mark for an answer that does not provide a logic argument. Every statement in your argument has to be in symbolic form. Please indicate the inference rule you are using in each inference step, and for each statement in the argument, indicate whether it is a statement given or a statement deduced in the inference