Please solve the following discrete mathematics questions.

1. Write the contrapositive of the following conditional statement as an English sentence. For you to win the contest it is necessary that you have the only winning ticket." 2. Let L(x,y) be the statement x loves y," where the domain of x and y consists of all people. Express the following sentence in terms of L(x,y), quantifiers, and logical operators. "There is exactly one person whom everybody loves." 3. Write the following argument in argument form (i.e., using predicates, quantifiers, and logical operators). "All rodents gnaw their food. Mice are rodents. Rabbits do not gnaw their good. Therefore, mice gnaw their food and rabbits are not rodents." 4. Write the assumption (i.e., what we suppose) in a proof by contradiction of the following statement. "At least 10 of any 64 days chosen must fall on the same day of the week