# Denominator negative, numerator negative but sign of fraction not positive?

I am trying to determine the sign of a fraction by determining the sign of the denominator and the numerator seperately using some parametric assumptions (ass6):

Assuming[ass6, TrueQ[Simplify[Denominator[detHesse] < 0]]]


which returns True and

Assuming[ass6, TrueQ[Simplify[Numerator[detHesse] < 0]]]


also returning True. However,

Assuming[ass6, TrueQ[Simplify[detHesse > 0]]]


returns False.

What am I missing? Thank you guys

• For a full answer can you post a self-contained code? Need to have the ass6 and detHesse. or just ass6 would probably be enough. Sep 9 '16 at 13:19
• Have you tried using FullSimplify rather than Simplify? Sep 9 '16 at 15:26
• Thank you @VahagnTumanyan and @BobHanlon. As @mikado pointed out the problem is that Simplify[detHesse > 0] doesn't simplify to Trueexplicitly, so that TrueQreturns False. I realized that the problem arises because I use Reduce in the definition of ass6. Sep 11 '16 at 10:51

From the help

Unless expr is manifestly True, TrueQ[expr] effectively assumes that expr is False.

So expressions that do not simplify to True will not necessarily return true even if they are.

eqns = {Cos[x]^2 + Sin[x]^2 == 1, x Gamma[x] == Gamma[x + 1]};

TrueQ /@ eqns
(* {False, False} *)

TrueQ[Simplify[#]] & /@ eqns
(* {True, False} *)

TrueQ[FullSimplify[#]] & /@ eqns
(* {True, True} *)


This behaviour is common to functions whose name ends in Q

• Thanks @mikado, that seems to be exactly the issue. In fact, I realized that the problem in my case seems to be that I use Reduce on my set of assumptions which suggests that Reduce does not what I thought it does. Sep 11 '16 at 10:52