If f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) and if f'(x) > 0 for every x ∈ (a, b), then f is increasing on [a, b].
I was giving the logical statement above and asked to negate it. I know its a conditional statement which means its negation is p∧¬q, but there are logical and's in the "p" part of the statement. Do I have to negate those and's as well?
My answer, which was wrong, was as follows
If there exists a function f that is either continuous or differentiable on the interval I or if f'(x) > 0 for every x ∈ (a, b) and f is not increasing on I.
The issue with your answer:
Let, A be "f is continuous on a closed interval [a, b]"
B be "differentiable on the open interval (a, b)"
C be "f'(x) > 0 for every x ∈ (a, b)"
D be "f is increasing on [a, b]."
The statement says:
$$(A \wedge B \wedge C) \rightarrow D$$
It's negation is:
$$(A \wedge B \wedge C) \wedge \neg D$$
Negation:
f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) and if f'(x) > 0 for every x ∈ (a, b) and f is not increasing on [a, b].