I have a problem with interpreting the results of my notebook outputs. It is important that I do not err with the interpretation.
Here is the full input:
$Assumptions =
0 < a <= 1/2 && 0 <= T < 1 && 0 <= d <= 1 && F > 0 && 0 < r < 1 &&
c >= 0 && a ∈ Reals && T ∈ Reals &&
d ∈ Reals && F ∈ Reals && r ∈ Reals &&
m ∈ Reals && c ∈ Reals &&
m > c && (-2 c + m - 2 F r)/(1 + d (-1 + r)) >= 0 &&
T*F < r*F + (1 - r)*d*T*F && m > 2*F; LogicalExpand@
Reduce[(-m - 2*F*r +
2*F*T)/(2*(-1 + r)*(-c + m -
F*T)) - ((1/(2*F*(-1 + r)^2))*(m + c*(-1 + r) +
F*(-1 + r)^2 - m*r +
Sqrt[((-1 + r)^2)*(c^2 + m^2 - 2*c*(F + m) + 2*c*F*r +
F*(F - 2*(F + m)*r + F*r^2))])) == 0 && $Assumptions]
(* False *)
$Assumptions =
0 < a <= 1/2 && 0 <= T < 1 && 0 <= d <= 1 && F > 0 && 0 < r < 1 &&
c >= 0 && a ∈ Reals && T ∈ Reals &&
d ∈ Reals && F ∈ Reals && r ∈ Reals &&
m ∈ Reals && c ∈ Reals &&
m > c && (-2 c + m - 2 F r)/(1 + d (-1 + r)) >= 0 &&
T*F < r*F + (1 - r)*d*T*F && m > 2*F; LogicalExpand@
Reduce[(-m - 2*F*r +
2*F*T)/(2*(-1 + r)*(-c + m -
F*T)) - ((1/(2*F*(-1 + r)^2))*(m + c*(-1 + r) +
F*(-1 + r)^2 - m*r +
Sqrt[((-1 + r)^2)*(c^2 + m^2 - 2*c*(F + m) + 2*c*F*r +
F*(F - 2*(F + m)*r + F*r^2))])) > 0 && $Assumptions]
(* False *)
$Assumptions =
0 < a <= 1/2 && 0 <= T < 1 && 0 <= d <= 1 && F > 0 && 0 < r < 1 &&
c >= 0 && a ∈ Reals && T ∈ Reals &&
d ∈ Reals && F ∈ Reals && r ∈ Reals &&
m ∈ Reals && c ∈ Reals &&
m > c && (-2 c + m - 2 F r)/(1 + d (-1 + r)) >= 0 &&
T*F < r*F + (1 - r)*d*T*F && m > 2*F; LogicalExpand@
Reduce[(-m - 2*F*r +
2*F*T)/(2*(-1 + r)*(-c + m -
F*T)) - ((1/(2*F*(-1 + r)^2))*(m + c*(-1 + r) +
F*(-1 + r)^2 - m*r +
Sqrt[((-1 + r)^2)*(c^2 + m^2 - 2*c*(F + m) + 2*c*F*r +
F*(F - 2*(F + m)*r + F*r^2))])) < 0 && $Assumptions]
(* (c == 0 && T == r && m > 0 && 0 < a && 0 < d && 0 < F &&
0 < r && F < m/2 && r < 1 && a <= 1/2 && d <= 1) || (c == 0 &&
m > 0 && 0 < a && 0 < F && 0 < r && F < m/2 && r < 1 && r < T &&
T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1) || (c == 0 &&
m > 0 && 0 < a && 0 < F && 0 < r && F < m/2 && r < 1 && T < r &&
0 <= d && 0 <= T && a <= 1/2 && d <= 1) || (T == r && m > 0 &&
0 < a && 0 < c && 0 < d && 0 < F && 0 < r && c < m/2 && r < 1 &&
a <= 1/2 && d <= 1 && F <= 1/2 (-2 c + m)) || (T == r && m > 0 &&
0 < a && 0 < c && 0 < d && 0 < r && c < m/2 && F < m/2 &&
1/2 (-2 c + m) < F && a <= 1/2 && d <= 1 &&
r <= (-2 c + m)/(2 F)) || (m > 0 && 0 < a && 0 < c && 0 < F &&
0 < r && c < m/2 && r < 1 && r < T &&
T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1 &&
F <= 1/2 (-2 c + m)) || (m > 0 && 0 < a && 0 < c && 0 < F &&
0 < r && c < m/2 && r < 1 && T < r && 0 <= d && 0 <= T &&
a <= 1/2 && d <= 1 && F <= 1/2 (-2 c + m)) || (m > 0 && 0 < a &&
0 < c && 0 < r && c < m/2 && F < m/2 && 1/2 (-2 c + m) < F &&
r < T && T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1 &&
r <= (-2 c + m)/(2 F)) || (m > 0 && 0 < a && 0 < c && 0 < r &&
c < m/2 && F < m/2 && 1/2 (-2 c + m) < F && T < r && 0 <= d &&
0 <= T && a <= 1/2 && d <= 1 && r <= (-2 c + m)/(2 F)) *)
What I need to know is if the researched term is positive, negative or zero. As you can see here, Mathematica tells me that the term is neither positive nor zero (False/False). However, it gives me many combinations of parameter assumptions for which the term gets negative. Why is this? If it is neither positive nor zero for any combinations shouldn't it always be negative? Why does it give me these combinations of assumptions that seem to be necessary. It is important for me to interpret the results correctly.
I always saw it like that: If I put in a term (not that specific term) and wanted to evaluate if it is for example negative, than Mathematica would give me all combinations of parameter assumptions -possible under the frame of my model- for which the term is negative. All possible cases would be departed by the double vertical lines. For this example:
(* (c == 0 && T == r && m > 0 && 0 < a && 0 < d && 0 < F &&
0 < r && F < m/2 && r < 1 && a <= 1/2 && d <= 1) || (c == 0 &&
m > 0 && 0 < a && 0 < F && 0 < r && F < m/2 && r < 1 && r < T &&
T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1) || (c == 0 &&
m > 0 && 0 < a && 0 < F && 0 < r && F < m/2 && r < 1 && T < r &&
0 <= d && 0 <= T && a <= 1/2 && d <= 1) || (T == r && m > 0 &&
0 < a && 0 < c && 0 < d && 0 < F && 0 < r && c < m/2 && r < 1 &&
a <= 1/2 && d <= 1 && F <= 1/2 (-2 c + m)) || (T == r && m > 0 &&
0 < a && 0 < c && 0 < d && 0 < r && c < m/2 && F < m/2 &&
1/2 (-2 c + m) < F && a <= 1/2 && d <= 1 &&
r <= (-2 c + m)/(2 F)) || (m > 0 && 0 < a && 0 < c && 0 < F &&
0 < r && c < m/2 && r < 1 && r < T &&
T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1 &&
F <= 1/2 (-2 c + m)) || (m > 0 && 0 < a && 0 < c && 0 < F &&
0 < r && c < m/2 && r < 1 && T < r && 0 <= d && 0 <= T &&
a <= 1/2 && d <= 1 && F <= 1/2 (-2 c + m)) || (m > 0 && 0 < a &&
0 < c && 0 < r && c < m/2 && F < m/2 && 1/2 (-2 c + m) < F &&
r < T && T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1 &&
r <= (-2 c + m)/(2 F)) || (m > 0 && 0 < a && 0 < c && 0 < r &&
c < m/2 && F < m/2 && 1/2 (-2 c + m) < F && T < r && 0 <= d &&
0 <= T && a <= 1/2 && d <= 1 && r <= (-2 c + m)/(2 F)) *)
would mean that the term is only negative if either the combination of assumptions
(c == 0 && T == r && m > 0 && 0 < a && 0 < d && 0 < F &&
0 < r && F < m/2 && r < 1 && a <= 1/2 && d <= 1)
or the combination
(c == 0 &&
m > 0 && 0 < a && 0 < F && 0 < r && F < m/2 && r < 1 && r < T &&
T < 1 && (r - T)/(-T + r T) < d && a <= 1/2 && d <= 1)
or the combination
(c == 0 &&
m > 0 && 0 < a && 0 < F && 0 < r && F < m/2 && r < 1 && T < r &&
0 <= d && 0 <= T && a <= 1/2 && d <= 1)
is fulfilled and so on and so forth. But is not negative when none of all stated combinations is fulfilled. Am I right with how I interpret this output? It is very important for my further research because when I don't interpret that right all my results that base on that are maybe meaningless.
I hope you can help me and I'm very grateful for your help.