1. 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

2. Refer to the statements of predicate calculus you provided for problem 1. Write the negation of each of those statements.

3. Give counterexamples to show each of the following is false:

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

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

5. Refer to the statements in problem 4 and indicate whether each is true or false. Provide your reasoning.