My understanding is that Reduce
gives all conditions (using or) where the input is true.
Now, $ \sqrt{xy} = \sqrt x \sqrt y $ , where $ x,y$ are real, under the following three conditions/cases
$ $
\begin{align*}
x\geq 0,y\geq0\
x\geq0,y\leq0\
x\leq0,y\geq 0 \
\end{align*}
$ $
but not when $ x<0,y<0$
This is verified by doing
ClearAll[x,y]
Assuming[Element[{x,y},Reals]&&x>= 0&&y<= 0,Simplify[ Sqrt[x*y] - Sqrt[x]*Sqrt[y]]]
Assuming[Element[{x,y},Reals]&&x<= 0&&y>= 0,Simplify[ Sqrt[x*y] - Sqrt[x]*Sqrt[y]]]
Assuming[Element[{x,y},Reals]&&x<= 0&&y>= 0,Simplify[ Sqrt[x*y] - Sqrt[x]*Sqrt[y]]]
Assuming[Element[{x,y},Reals]&&x<= 0&&y<= 0,Simplify[ Sqrt[x*y] - Sqrt[x]*Sqrt[y]]]
Then why does
Reduce[ Sqrt[x*y] - Sqrt[x]*Sqrt[y]==0,{x,y},Reals]
Give only one of the 3 cases above?
Is my understanding of Reduce
wrong or should Reduce
have given the other two cases?
V 12 on windows.