Write your proofs clearly using complete sentences. Try silently reading what you 've written and decide if any words are missing. If you introduce new notation or variable names, state explicitly what they represent. Don't use any symbols such as => or V in your proofs; use words instead. It is allowed and encouraged to work with your classmates on these problems, but your submission must be entirely your own work. Solutions which are copied from a student or another source will receive a score of 0. 1. Suppose that \"Girls like cats\" and \"Boys like dogs\" are true statements and that \"Everybody likes horses\" is a false statement. Which of the following are true? Hint: Label each of the given statements, and think about each of the following using connec- tives. a. If girls like cats, then everybody likes horses. b. If everybody likes horses. then boys like dogs. c. If everybody likes horses, or girls like cats, then boys like dogs. d. If boys like dogs and girls like cats, then everybody likes horses. 2. In the following exercises, justify your answers using truth tables. a. Are P V (Q A R) and (P V Q) A R logically equivalent? b. Is any of the two statements from part (a) logically equivalent to (P A R) V (Q A R)? 3. In the following exercises, justify your answers using truth tables. a. Is ((-IP A Q) V (P A IQ)) {it -I(P {I} Q) a tautology? b. Is [(P V Q) A (-IP)] A (-IQ) is a contradiction? 4. Suppose that the statement \"If it snows in March, then no tree will bloom in April\" is a true statement. a. We see a tree blooming in April. What can we conclude (if anything?) b. It did not snow in March. What can we conclude (if anything?) 5. Rewrite each of the following as an if-then statement. a. In order for n to be even, it is sufcient that n2 is even. b. In order for n to be even, it is necessary that n2 is even. c. A function is differentiable only if it is continuous. (1. None of the integers from 200 to 210 (inclusive) is prime. 6. Write the negations of each of the following statements. Avoid simply adding the word \"not\" to the statement. a. The temperature in Irvine is always at least 40 degrees. b. Wehavea. 5 2andb > 7. c. There are exactly two solutions to the equation f(:r:) = 0. d. If Carol and Debbie are siblings, then Edward is their dad. 7. Assume f (:5) is a continuous function whose domain is all real numbers. Consider the statement, a Iff(2) 0,thenf(a:) = 0forsome2