Translate the following English statements into predicate logic formulae. The domain is the set of integers. Use the following predicate symbols, function symbols and constant symbols.
- Prime(x) iff x is prime
- Greater(x, y) iff x > y
- Even(x) iff x is even
- Equals(x,y) iff x=y
- sum(x, y), a function that returns x + y
- 0,1,2,the constant symbols with their usual meanings
I tried them, but don't know if they're correct.
(a) The relation Greater is transitive.
(∀x(∃y(∃z (Greater(x,y) ∧ Greater(y,z) -> Greater(x,z))))
(b) Every even number is the sum of two odd numbers. (Use (¬Even(x)) to express that x is an odd number.)
(c) All primes except 2 are odd.
(∀x(Prime(x) ∧ ¬(Equals(x,2) -> ¬Even(x))
(d) There are an infinite number of primes that differ by 2. (The Prime Pair Conjecture)
(∀x(∃y (Prime(x) ∧ Equals(y,sum(x,2)) ∧ Prime(y)))) From what I remember, we aren't suppose to put predicate symbol (sum(x,2)) inside Equals. How do I do this?
(e) For every positive integer, n, there is a prime number between n and 2n. (Bertrand's Postulate) (Note that you do not have multiplication, but you can get by without it.)
(∀x(∃y (Greater(x,1) -> (Greater(x,y) ∧ Prime(y) ^ Greater(y, Sum(x,x))))) -Same problem as d.