My understanding is that `Reduce`

gives all conditions (using or) where the entry is true.

Now, $ sqrt {xy} = sqrt x sqrt and $, where $ x, y $ are **real**, under the following three conditions / cases

$$

begin {align *}

x geq 0, y geq0 \

x geq0, and leq0 \

x leq0, y geq 0 \

end {align *}

$$

but not when $ x <0, and <0 $

This is verified by doing

```
Clear all[x,y]
Assuming[Element[{x,y},Reals]&& x> = 0 && and <= 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 && and <= 0, simplify[ Sqrt[x*y] - Sqrt[x]* Sqrt[y]]]
```

Then because it does

```
Reduce[ Sqrt[x*y] - Sqrt[x]* Sqrt[y]== 0, {x, y}, Real]
```

Give only one of the 3 previous cases?

It is my understanding of `Reduce`

wrong or should `Reduce`

Have you given the other two cases?

V 12 in the windows.