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We have this predicate:

Prime(x) ≡ x>1 ∧(∀y)(Divide(y,x) → (y=1 ∨ y=x)).

Now I have to turn this into an English sentence. I've come up with three answers:

  1. x is a prime number if x is greater than 1, and if for every given y that is divisible by x, then y is either 1 or x itself.

  2. x is a prime number if x is greater than 1, and for every given y, if y is divisible by x, then y is either 1 or x itself.

  3. x is a prime number if x is greater than 1, and if for every given y, if y is divisible by x, then y is either 1 or x itself.

Which one is correct and if none are correct, how should I write it?

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Here is the predicate under consideration:

Prime(x) ≡ x>1 ∧(∀y)(Divide(y,x) → (y=1 ∨ y=x)).

The goal is to write this as an English sentence. The Divide relation seems to suggest that y divides x, not that y is divisible by x. It might be represented in English as

The natural number x is a prime number if and only if x is greater than 1 and for all natural numbers y if y divides x then y = 1 or y = x.

As a check on primality, here is Wikipedia's definition of prime number:

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.


Wikipedia contributors. (2019, June 7). Prime number. In Wikipedia, The Free Encyclopedia. Retrieved 04:32, June 16, 2019, from https://en.wikipedia.org/w/index.php?title=Prime_number&oldid=900751049

  • thank you very much. I just had another question: I have seen that the notion ≡ is sometimes used as "if and only of". but I should not use it in this sentence. right? – Daruis soli Jun 16 at 10:08
  • "if and only if" – Daruis soli Jun 16 at 10:49
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    @Daruissoli Yes, I agree, I should have put "if and only if" there instead of "if". I will make the edit to change that. – Frank Hubeny Jun 16 at 12:13

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