Express the definition of prime in the language of quantifiers. Then state the negation in quantities, and interpret it in your own words

Question

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?

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

Thanks

Answer 1

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:

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