The negation of an implication.

I have the following statement and I'm not sure what the negation is. The statement is:

If $F:\mathbb{R}\to\mathbb{R}$ is a function satisfying some regularity assumptions $(R1)$ then we have $$\lim_{x\to\infty}F(x)\ge 0.$$

I think the negation should be:

If $F:\mathbb{R}\to\mathbb{R}$ is a function satisfying some regularity assumptions $(R1)$ then we have $$\lim_{x\to\infty}F(x)< 0.$$

Is this correct?

Solution 1:

Recall that $p\rightarrow q$ is equivalent to $\lnot p\lor q$. Therefore the negation of the implication is the same as negating the disjunction. Using DeMorgan laws we have: $$\lnot(\lnot p\lor q)\equiv\lnot\lnot p\land\lnot q\equiv p\land\lnot q.$$

Therefore the negation of "If one then two" is "one and not two".