Discrete Structures

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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:
For all positive integers x and y, xy > x + y
For all real numbers x, x > 1/x
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.