Show all steps. Explain everything. 1. Below is the Goldbach's conjecture, which is not known to be true and not known to be false: Every positive even number can be expressed as the sum of two prime numbers. a. Rewrite the Goldbach's conjecture formally. b. Negate it formally and then informally. c. Rewrite the conjecture as a universal conditional statement first informally then formally. d. Write its converse informally and formally e. Write its inverse informally and formally f. Write its contrapositive informally and formally g. Negate the conditional statement informally and formally 2. Below are several formal statements. a-c Rewrite the statements informally, avoid using symbols. Use English as much as possible. parallelograms, a. VreR, reP where R=set of rectangles, P-set b. 3A: A has 20 elements where A=set of athletes c. VreQ,3a,beZ such that r = a d. Negate: VreQ,3a,beZ such that r = -formally. e. Reduce to a universal conditional statement formally: VxeR, Vx < 2, 2 3. What conclusion can be drawn from the following? Use symbolic logic. Explain. a. If it is raining, then Prof. Feinman will go home. b. If it is morning, then Prof. Feinman will not go home. c. If it is raining, then it is morning. 4. Given: "If P is a rose, than it has at least 15 petals" is true. Which of the following is (are) true which is (are) false? Explain. a. If it B is not a book, then it has less than 10 pages. b. B has at least 10 pages only if it is a book. c. If there are at least 10 pages, then B is a book. d. If B has less than 10 petals, then it is not a book. e. A necessary condition for B to have at least 10 pages is that it is a book. f. A sufficient condition for B to have at least 10 pages is that it is book. 5. Prove or disprove the given argument by listing all logic steps, stating and naming the used laws. Note: "+" =if and only if. Given: a) a b b) -e v d c) c +f d) (d f) a e) ~d V c f) Therefore, (f Ab) + d 6. a. By using direct proof, prove that all squared even numbers are divisible by 4. b. Prove or disprove: the product of odd numbers, is an odd number.