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Thread: Determining truth value of statement

  1. #1
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    Question 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.

    This question is for homework so please only give helpful hints!

    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.
    Last edited by otownsend; May 15th 2019 at 02:52 PM.
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  2. #2
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    Re: Determining truth value of statement

    Quote Originally Posted by otownsend View Post
    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.
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