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A. Solve and submit your solutions in a paper the following set of exercises: 1. The conclusion is a type of proposition P. where P is the proposition "Q begets R" or "Q is sufficient for R" or "R is necessary when Q" or, in notation format, P: Q => R. Truth table is a table where the values of the proposition are evaluated based on the constituent propositions. For P, the truth table based on the logical values of Q and R is the following: (Value of ) Q (Value of) R (Resulting value of) P T T T F F F T Conceive your own propositions and verify (prove to yourselves) the truth table above. In particular, the third row of it where the reasoning of a True proposition from a False is a Valid deduction. 2. Let's consider the outcomes of the throwing of two dices. Model the result of each throw as an element of one of the sets: Al = {1, 2, 3, 4, 5, 6) and A2 = (1, 2, 3, 4, 5, 6). So, each throw can be represented as an element of the set T = ((x, y): Vx e A1 A Vy E A2). a) Write down the set T in roster notation. b) Consider the sets i) B= {(x, y): (x, y) = TA x + y = 7) and ii) C = {(x, y): (x, y) ETA (x = 6 Vy = 6). Write down the sets B and C in roster notation and Venn diagrams. c) Evaluate the intersection B / C, the union B U C, the (absolute) complement of the union B U C, (B U C) d) Provide description of the latter sets in in set builder notation. e) Provide a Venn diagram for the B / C 3. Simplify/solve the expressions (where D is the last digit of your student ID:4204219 if it is equal to 0, use D=10) (8 - 3[-2(2 - 5) - 4(D - 6)1] - (8 - 3[-2(5 - 7) -5(4 - D-3*5(3+6) + (5-D)=(5-D)+2+4 2)]) 7+3-6+6 89-3*5*5 3 D$2-3*3 D+3+4 *10 [3*3-(-2)]*[3*3+(-2)] + (12-2*3)*(12+2*3) = (D: - 8) 3 9 + 20 104. i) A not loaded truck weighs 2.58 tons (1 ton = 1000 Kg). It is loaded with a number of boxes, each of which weighs 56 Kgs. The truck now weighs 6780 Kgs. How many boxes were loaded onto the truck? Model the problem and then solve it. (Full marks: 10 points) ii) Grandma Kate decided to give /$/! 110+20 of her money to her three grandchildren. Maria would receive 10+D more than John who would receive 15+D less than Alexandra. How much would each child get? Model the problem and solve it. (Full marks: 10 points) iii) For the numbers 168 and 196 a) find their factors, b) their GCD and c) their LCM (Full marks: 10 points) 5. i) Write down the division formula for: a. 424:3, b. 1517 : 15 ii) A natural number when divided by 3 gives a quotient of 237 and a remainder of 1. Which is this number? iii) If today is a Tuesday what is the day after 136+D days (where D is the last digit of your student ID.) student ID:4204219