I'm trying to wrap my head around logical implications, equivalences, etc. and I made up two random statements. This is my attempt at proving one implies the other:
Assume Statement $A$ and the antecedent of Statement $B$
Dividing Statement $B$ into cases:
Case 1: $z<x<y$
From $A$ we can conlude that $0<f(z)<f(x)<f(y)$, meaning $f(x)+f(y)+f(z)>0$
Case 2: $x<z<y$ and Case 3: $x<y<z$
(Same reasoning as in case 1)
By showing that all three possible cases are true given the assumption that $A$ is true, can I say that Statement $A$ implies Statement $B$?
And if possible, how would you prove that $B$ implies $A$?