Question

Hi everyone:

I am not sure about the following thing I did. Let J be a countable finite set, and f0jk and f1jk be two continuous functions defined on [0,1]. Consider the following statement:

∀lj∈J,∀x∈[0,1],f0jk(x)≤f1jk(x)

Negating the above statement gives me:

∃lj∈J,∃xˆ∈[0,1],f0jk(xˆ)>f1jk(xˆ)

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?

Answer 1

yes, you negate for all... for all ... with
there exists... there exists... not whatever

Answer 2

and so yes, all you have to do is come up with one that violates the condition

Answer 3

thanks a lot!