Really a Question about Notation

[Atomised]

Homework Statement

Form negation and then either prove statement or negation:

$\forall$y $\in$ {$x | x \in Z, x>=1$}, $5y^2+5y+1$ is a prime number.

The Attempt at a Solution

Answer given:

$\exists$y $\in$ {$x | x \in Z, x>=1$} such that $5y^2+5y+1$ is not prime.

The negation is true, counterexample follows.

My question is why are $x,y$ mixed up?

 

[electricspit]

What do you mean why are $x, y$ mixed up?

 

[Atomised]

Well, there exists y belonging to a set where the elements are defined by x...

Would it be valid to express it as there exists x belonging to a set where the elements are defined by x... or the above the preferred way of doing it?

 

[electricspit]

The way I interpret it is like this:

Let $X$ be defined as the set of numbers $\{x|x\in\mathbb{Z}, x\geq1\}$.

Then:

$\exists y \in X \text{ such that } 5y^2+5y+1 \text{ is not prime}$

This might make it clearer that the set is being defined by the dummy variable $x$. It has no real purpose besides just being a dummy variable from what I can see.

 

[Atomised]

Very helpful thank you.

 

[Ray Vickson]

Homework Statement

Form negation and then either prove statement or negation:

$\forall$y $\in$ {$x | x \in Z, x>=1$}, $5y^2+5y+1$ is a prime number.

The Attempt at a Solution

Answer given:

$\exists$y $\in$ {$x | x \in Z, x>=1$} such that $5y^2+5y+1$ is not prime.

The negation is true, counterexample follows.

My question is why are $x,y$ mixed up?

In plain English, the statement is: $5 y^2 + 5y+1$ is prime for any positive integer $y$. Of course, as you have shown, the statement is false.