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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.

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  • (a) you want to universally quantify all three variables. Transitivity imposes a constraints on all triples of elements.

  • (b) You want to say that for every number that is even there are two numbers that are odd and such that their sum equals the given number.

  • (c) is correct. (Assuming that conjunction binds more strongly than implication.)

  • (d) What you say there is that every natural number is prime and has a twin. That's not the twin-prime conjecture. Note that "sum" is a function symbol, not a predicate symbol.

  • (e) You should switch the arguments to "Greater."

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For d) and e): $sum$ is a function that denotes an object (the return value of the function), and therefore it can be used as an argument inside a predicate just fine!

However, as Fabio pointed out, your symbolization for d is not correct. To get an infinite number of twin primes, you can say that there is at least one twin prime, and that for every twin prime there is a larger twin prime.

Your e )is almost correct, but you need to use $Greater(y,x)$ and $Greater(sum(x,x),y)$. And if you really want to capture Bertrand's postulate (for any $x \ge 1$ there is a prime $y$ such that $x < y \le 2x$) you need to use $(Greater(sum(x,x),y) \lor Equals(sum(x,x),y))$ at the end.

Also: Careful with those parentheses! for example, c) should be $\forall x ((Prime(x) \land \neg Equals(x,2)) \rightarrow \neg Even(x))$

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Bram has pointed out that in $(e)$ you need to add $Equals(sum(x,x),y)$ to the consequent but to make this translation the perfect one you also have to add $Equals(x,1)$ to the antecedent to capture the $x \ge 1$ part of the Bertrand's Postulate.

Lets look at $(d)$ now. For clearance I'm using slightly different notation for atomic formulas $$ \begin{array}{} P(x) & \text{for Prime (x)} \\ E(x,y) & \text{for Equals (x, y)} \\ G(x,y) & \text{for Greater (x, y)} \end{array} $$ Now we can translate $(d)$ into the following formula:

$$\forall x ((P(x) \land \exists y(P(y) \land E(y, sum(x, 2))) \to \exists z(P(z) \land G(z, x) \land \exists w(P(w) \land E(w, sum(z, 2)))))$$

It says that, for every $x$ which is $prime$ and has a $twin\ prime$, which is greater than $x$ by $2$, we can find a $prime\ z$ which is greater than $x$ and also have a $twin\ prime$.

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