1. Translate the following English sentences into statements of predicate calculus. (20 points, 4 parts 5 points each)

a. All programmers enjoy discrete mathematics

b. Some integers are not odd

c. Every integer that is divisible by 2 is even

d. There exists a natural number that is not a positive integer

2. Refer to the statements of predicate calculus you provided for problem 1. Write the negation of each of those statements. . (20 points, 4 parts 5 points each)

3. Give counterexamples to show each of the following is false: (20 points, 2 parts 10 points each)

a. For all positive integers x and y, xy > x + y

b. For all real numbers x, x > 1/x

4. Translate the following English sentences into statements of predicate calculus that contain double quantifiers: (20 points, 2 parts 10 points each) a. Any even integer is equal to twice some other even integer b. Some natural number is no bigger than every natural number

5. Refer to the statements in problem 4 and indicate whether each is true or false. Provide your reasoning. (20 points, 2 parts 10 points each)