- Translate the following English sentences into statements of predicate calculus.
- All programmers enjoy discrete mathematics
- Some integers are not odd
- Every integer that is divisible by 2 is even
- There exists a natural number that is not a positive integer
- Refer to the statements of predicate calculus you provided for problem 1. Write the negation of each of those statements.
- Give counterexamples to show each of the following is false:

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

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

4. Translate the following English sentences into statements of predicate calculus that contain double quantifiers:

- Any even integer is equal to twice some other even integer
- Some natural number is no bigger than every natural number
- Refer to the statements in problem 4 and indicate whether each is true or false. Provide your reasoning.