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Here is my draft of Answer.

Suppose p is a prime number, P€N and p not equal to1, and for all a€N, for all b € N, if P=a•b then a =1 or b=1

My questions: 1. Do I express the definition of prime in the language of quantifiers correctly? And how to show “ and “ “ if” in quantifiers type?

  1. How do we negate this? Just add a negation mark? And change “ for all” to “there exists”?

Thanks

Leir Leir
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1 Answers1

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Don't start with: "suppose $p$ is a prime number ..." You are not giving a proof, but rather stating a definition.

So, start with "$p$ is a prime number if and only if ..."

And yes, you got all the conditions correct: "$p$ is a prime number iff $p \in \mathbb{N}$ and $p \not = 1$ and for all $a \in \mathbb{N}$ and $b \in \mathbb{N}$: if $p = a \cdot b$ then $a=1$ or $b=1$"

In first-order logic, this can be translated as:

$$\forall p ( Prime(p) \leftrightarrow (p \in \mathbb{N} \land p \not = 1\land \forall a \forall b ((a \in \mathbb{N} \land b \in \mathbb{N} \land p = a \cdot b) \rightarrow (a=1 \lor b=1))))$$

Bram28
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  • Thanks, and how do we state the negation in quantifiers? Here is my draft of answer: – Leir Leir Apr 23 '18 at 23:46
  • “ not (there exists prime(p) <—> not(p€N or p not equal to 1 and there exists a and b (( a €N and b €n and p=a•b) <—> not(a=1 or b=1)))). – Leir Leir Apr 23 '18 at 23:51
  • Sorry about my terrible typing, I just don’t know how to use those math marks. – Leir Leir Apr 23 '18 at 23:51
  • So here is a question, if I want to show the negation of this statement in the language of quantifiers, do I just need to add not before the curly braces, and change and to or? – Leir Leir Apr 23 '18 at 23:53