MATH 74 HOMEWORK 1DUE TUESDAY, SEPTEMBER 2ND(1) Do Problems 1.3, 1.4, and 1.5 in Eccles.(2) Convert the following sentences into the notation of propositionaland predicate logic, using the given list of predicates. The firstproblem is done as an example.Predicates: T (x) = “x is tired.” B(x) = “x goes to bed early.”C(x) = “x is cranky.” L(x, y) = “x likes y.” W (x, y) = “x wantsy.” E(x, y) = “x eats y.” R(x, y) = “x rocks y like a hurricane.”(a) If Steve is tired, then he goes to bed early.Solution: T (Steve) ⇒ B(Steve).(b) Steve is tired and cranky.(c) Steve only goes to bed early if he is tired.(d) Steve likes doughnuts and coffee.(e) Steve likes either doughnuts or coffee (possibly both!).(f) Steve likes either doughnuts or coffee, but not both.(g) Steve only likes coffee if he is cranky.(h) Steve eats doughnuts whenever he wants doughnuts.(i) If Steve eats doughnuts and plums, then Steve is tired andcranky and goes to bed early.(j) If Steve eats a doughnut, then it rocks him like a hurricane.(3) Convert the following sentences into the notation of propositionaland predicate logic, using the given list of predicates. You may usethe predicates =, <, >. The first problem is done as an example.Predicates: E(n) = “n is even.” P (n) = “n is prime.”(a) Every prime number is odd, except for 2.Solution: (P (n) and (not n = 2))⇒(not E(n)).(b) The only even prime number is 2.(c) The square of any nonzero numbe r is positive. [Positive meansgreater than 0, not greater than or equal to 0.](d) The square of any odd number is odd.(e) Every positive number is either equal to 1 or greater than 1.1MATH 74 HOMEWORK 1 DUE TUESDAY, SEPTEMBER 2ND 2(4) Let P (n) be the predicate “n is even” and let Q(n) be the predicate“n2is even.” For which integers n is P (n) ⇒ Q(n) true? For whichones is it false? Draw a truth table that illustrates this.(5) Let P (n, m) be the state me nt “If n is greater than m, then n2isgreater than m2.” For which integers n and m is P (n, m) true? Forwhich ones is it