Translate English sentences

• 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