Events that carry negative information

I want to answer the following exercise:

An event $F$ is said to carry negative information about an event $E$, and we write $F \downarrow E$ if $P(E|F) \le P(E)$

Prove or give counterexamples of the following assertions:

a) If $F \downarrow E$ then $E \downarrow F$

my attempt: we have that $\frac{P(EF)}{P(F)} \leq P(E)$ then we have that $P(FE)=P(EF) \leq P(E)P(F)$ therefore $\frac{P(FE)}{P(E)} \leq P(F)$

b)If $F \downarrow E$ and $E \downarrow G$ then $F \downarrow G$

c)If $F \downarrow E$ and $G \downarrow E$ then $FG \downarrow E$

Repaeat a),b),c) but with $\uparrow$(is the same definition but with greater than instead of lower than)

and my questions are, Am I right in my attempt?, I can't figure out how in b) I will get $P(GF)$, and if I am wrong how can I get the counterexamples for a),b),c) I think that the same way to proceed with $\downarrow$ is equal to $\uparrow$,then if I have the $\downarrow$ part the other one is equal only the inequalities are inverted,isn't?

Thanks a lot for your help, I appreciate it very much :)

Solution 1:

Hint(s):

For part (b), what happens if $G=F$?

For part (c), what if $F$ and $G$ are events where $F \downarrow G$, and $E$ is the event $$E = (F \text{ and } G) \text{ or } (\text{not }F \text{ and not } G)$$ (Try drawing a Venn Diagram if you're having trouble seeing what I'm describing. $F$ and $G$ should be two ovals with a small intersection relative to their sizes, and $E$ should be the area both outside the two ovals and inside their intersection).