Let me first state the problem:
Let us define F carries positive information for E if and only if $P(E| F) \geq P(E)$.
Prove or disprove these claims:
a. If $F \uparrow E$ and $E \uparrow G$, then $F \uparrow G$
b. If $F \uparrow$ $E$ and $G \uparrow E$ then $FG \uparrow E$
This problem makes me really stuck. There's another version of this question asked here, which contains the "negative-carry" version of this.
Back to the problem, I want to mimic the approach in the above link but can't. My attempt for b is to try $F$ and $F^c$, but failed. In fact, $F \uparrow E $ does not imply $F^c | E$. My temptation is to use $F \uparrow F^c \iff P(F) \leq 0$ to disprove b.
Any hint for me to work this out? I'm super stuck at this. :(