Have I negated the statement "for every prime number $p$, $p+7$ is composite" correctly? This is the original statement:

For every prime number $p$, $p+7$ is composite.

This is my negation:

There exists a prime number $p$, where $p+7$ is prime.

Have I negated this correctly?
 A: Yes it is.
If you want to get this is an "automatic way", you can write the sentence logically:
$$\forall p \in Prime, p+7 \in Comp.$$
The negation is $$\exists p\in Prime, p+7\notin Comp.$$
To negate a sentence, switch the quantifiers (but not their domain), and negate all predicates.
A: Strict negation: a prime number $p$ exists such that $p+7$ is not composite.
Note that $1$ is a number that is not prime and not composite. Of course (working in $\mathbb N$) from $p>1$ it follows that $p+7\neq 1$, but that is a side-issue.
A: Yes you are right.
The negation would be: there exists a prime number $p$ such that $p+7$ is not composite, or equivalently, there exists a prime number $p$ such that $p+7$ is prime.
A: The negation of $\forall x \in Y: P(x)$ is $\exists x \in Y: \lnot P(x)$.
Specifically, the negation of: $$\forall p \in \text{prime numbers} : p+7\ \text{is composite}$$ is: $$\exists p \in \text{prime numbers} : \lnot( p+7\ \text{is composite}),$$ which is equivalent to: $$\exists p \in \text{prime numbers} : p+7\ \text{is not composite},$$ or, in English, "there is a prime number $p$ for which $p+7$ is not composite."
Further, noting that:


*

*all prime numbers are integers greater than 1,

*if $p$ is an integer greater than 1, then $p+7$ is an integer greater than 1, and 

*an integer greater than 1 is prime if and only if it is not composite,


we can rephrase the statement equivalently as "there is a prime number $p$ for which $p+7$ is also prime."
