# Thread: Determining truth value of statement

1. ## Determining truth value of statement

Hi,

I am really hoping someone can help guide me in the right direction in terms of knowing whether the following statement is true or not.

Here is the statement: $\forall f \in F, \exists g \in F, \forall n \in \{1, 2, 3\}, \forall x \in \mathbb{R}, \exists y\in\mathbb{R}, x < y \wedge |g(y) - (f(y) + 2^n)| < \frac{1}{4}$

I am sensing that given $n$ could be either 1, 2, or 3, makes it so that any g would not work. If $n$ was defined before g in the order of quantifiers in the above statement then you could define $g(x) = f(y) + 2^n + 1/5$, which would make the statement be true, but this is clearly not the case. Is this the correct way to think about it? And if so, how could I come up with a counter-example?

Also, lets say F is the set of all functions.

2. ## Re: Determining truth value of statement

Originally Posted by otownsend
Here is the statement: $\forall f \in F, \exists g \in F, \forall n \in \{1, 2, 3\}, \forall x \in \mathbb{R}, \exists y\in\mathbb{R}, x < y \wedge |g(y) - (f(y) + 2^n)| < \frac{1}{4}$
You need to review and correct this post. I think that you dropped an x as in g(x) from the question.