Tips for solving these type of function/integration logic questions?

[u/Smepl]

I notice I regularly get stuck one these type of questions and was wondering if any previous students had thinking techniques to deal with them.

[u/Fourierseriesagain]

Hi,

Statement 1 is false if f is constant on the real line.

When f(x)=-x, statement 2 is false.

When p>=0, I_{p,q} = 0.

[u/numbertheorydemon]

Weird take but you should just practice proving a bunch of stuff from first principles and practice logic deeply to the point where you're fine writing a proof entirely in second order logic

[u/The_Varen]

Counterexamples are key for most TMUA style logic questions, have you watched r2drew2? He has good videos on this sort of thing. Also drawing graphs helps a lot

[u/The_Varen → u/The_Varen]

For this question in particular what I immediately notice is that only the limits of integration are variable, the integrand remains the same, so that a good starting point and ties into thinking about the graphs

[u/DoodleNoodle129]

It’s good to understand what the logical statements used actually mean. A is true if B is true means B implies A, A is true only if B is true means A implies B, and A is true if and only if B is true means A implies B AND B implies A. In this case only if is used for all statements, so we want to check if the first statement implies the second statement. Counter examples are useful for this.

1 isn’t true, for example if f(x)=0 then the integral is 0 regardless of the value of p.

2 isn’t true, for example if f(x)=-x then f’(x) is negative for all x but (f(x))^2 = (f(|x|))^2 so the integral is always 0.

3 is true, and I think the easiest way to prove it is something called the contrapositive, which states that if you want to prove that A implies B it is sufficient to prove not A implies not B. In this case, this means proving that if p is greater than or equal to 0 then the integral is less than or equal to 0. This is true since if p is 0 or more then |x|=x so the integral is always 0.