Using the domain of natural numbers N+ (i.e., 1, 2, 3, .), lets define D(x, y) to be true when y divides x and false otherwise. For example, if x=10 and y=2, then D(x, y) is true because 10 is divisible by 2. Contrary, if x=7 and y=2, then D(x, y) is false since 7 is not divisible by 2 over the domain of natural numbers. Now, lets define P(x) over N+ such that P(x) is true if x is a prime number and false otherwise.

a) Using quantifiers, logical operators, and D(x, y) as previously defined, write the symbolic representation to express P(x). Hint: We say that any natural number x is prime if the only numbers that divide it are 1 and x. *Note that 1 is not a prime number. Write a symbolic representation for a WFF that accurately defines P(x).

b) Consider the factual statement, There are an infinite number of natural numbers that are divisible by 6. Using quantifiers, logical operators, and our previously defined D(x, y), express this statement as a WFF (well-formed formula). You may continue to assume a domain of N+. Write a symbolic representation for a WFF that accurately defines the statement, There are an infinite number of natural numbers that are divisible by 6.

c) Consider a world in which the following is true -- There is NOT an infinite number of prime numbers. Using quantifiers, logical operators, and P(x) as previously defined, express this statement as a WFF (well-formed formula). You may continue to assume a domain of N+. Write a symbolic representation for a WFF that accurately defines the statement, There is NOT an infinite number of prime numbers.