Negating a Statement: Understanding the Rules and Implications

[cris(c)]

negating a statement...need help urgently!

Hi everyone:

I am not sure about the following thing I did. Let J be a countable finite set, and $f_{jk}^{0}$ and $f_{jk}^{1}$ be two continuous functions defined on [0,1]. Consider the following statement:

$\forall lj\in J,\forall x\in[0,1],\: \, f_{jk}^{0}(x)\leq f_{jk}^{1}(x)$

Negating the above statement gives me:

$\exists lj\in J,\exists \hat{x} \in[0,1],\: \, f_{jk}^{0}(\hat{x})> f_{jk} ^{1}(\hat{x})$

Question 1: Am I correct in the way I negate the original statement?

Question 2 (and perhaps the most important): The fact that the negation involves only one member gives the freedom to assume that every other element satisfies the properties in the original statement? i.e., can I assume, while constructing a proof, that $\forall hz\in J$ other than l and j, $f_{lz}^{0}(x)\leq f_{lz}^{1}(x)\: \forall x\in[0,1]$?

Thanks a lot! I truly appreciate any help you can give me!

 

[cris(c)]

any one out there willing to help?