Does applying Reduce result in an equivalence or a oneway implication or both?

Question

If I type something like this into Mathematica:

(*start*)
Clear[s, A, B, n, z, k];
n = 19;
Reduce[rho == s + 1/n + 1/(1 - A/B) &&
s + 1/n + 1/(1 - A/B) == Conjugate[-s - 1/n + 1/(1 - B/A)] &&
B != 0, Re[rho]]
(*end*)

Output:

((Im[rho] < Im[s] && 
     Re[rho] == 1/
      2 && (Re[B] < 0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
       Re[B] > 0)) || (Im[rho] == 
      Im[s] && ((Re[s] < 17/38 && 
         Re[rho] == 1/
          2 && (Re[B] < 
            0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
           Re[B] > 0)) || (Re[s] > 17/38 && 
         Re[rho] == 1/
          2 && (Re[B] < 
            0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
           Re[B] > 0)))) || (Im[rho] > Im[s] && 
     Re[rho] == 1/
      2 && (Re[B] < 0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
       Re[B] > 0))) && 
 A == (-20 B + 19 B rho - 19 B s)/(-1 + 19 rho - 19 s)

Is the resulting output equivalent to the statements in the program or is it a oneway implication or a mix of both? I am not strong in logic, and any hints are welcome.

I now found the answer in the Mathematica documentation for Reduce: "The result of Reduce[expr,vars] always describes exactly the same mathematical set as expr." — Mats Granvik, Feb 14 '23 at 13:06

Mats Granvik · Answer 1 · 2023-02-13T18:18:18.723

After posting the question I found this:

(* start *)
Clear[s, A, B, n, z, k];
n = 19;
eq1 = rho == s + 1/n + 1/(1 - A/B) && 
  s + 1/n + 1/(1 - A/B) == Conjugate[-s - 1/n + 1/(1 - B/A)] && B != 0
TreeForm[eq1]
(* end *)

And the output:

(*start*)
Clear[s, A, B, n, z, k];
eq2 =
  ((Im[rho] < Im[s] && 
       Re[rho] == 
        1/2 && (Re[B] < 
          0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
         Re[B] > 0)) || (Im[rho] == 
        Im[s] && ((Re[s] < 17/38 && 
           Re[rho] == 
            1/2 && (Re[B] < 
              0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
             Re[B] > 0)) || (Re[s] > 17/38 && 
           Re[rho] == 
            1/2 && (Re[B] < 
              0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
             Re[B] > 0)))) || (Im[rho] > Im[s] && 
       Re[rho] == 
        1/2 && (Re[B] < 
          0 || (Re[B] == 0 && (Im[B] < 0 || Im[B] > 0)) || 
         Re[B] > 0))) && 
   A == (-20 B + 19 B rho - 19 B s)/(-1 + 19 rho - 19 s);
TreeForm[eq2]
(*end*)

Does applying Reduce result in an equivalence or a oneway implication or both?

1 Answers1