• Home
  • About Us
  • Contact Us
  • Privacy Policy
  • Special Offers
Business Intelligence Info
  • Business Intelligence
    • BI News and Info
    • Big Data
    • Mobile and Cloud
    • Self-Service BI
  • CRM
    • CRM News and Info
    • InfusionSoft
    • Microsoft Dynamics CRM
    • NetSuite
    • OnContact
    • Salesforce
    • Workbooks
  • Data Mining
    • Pentaho
    • Sisense
    • Tableau
    • TIBCO Spotfire
  • Data Warehousing
    • DWH News and Info
    • IBM DB2
    • Microsoft SQL Server
    • Oracle
    • Teradata
  • Predictive Analytics
    • FICO
    • KNIME
    • Mathematica
    • Matlab
    • Minitab
    • RapidMiner
    • Revolution
    • SAP
    • SAS/SPSS
  • Humor

Should Reduce give all cases when $\sqrt{x y} = \sqrt x \sqrt y$?

July 20, 2019   BI News and Info

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]]]

dj9xa Should Reduce give all cases when $\sqrt{x y} = \sqrt x \sqrt y$?

Then why does

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

Give only one of the 3 cases above?

kNd60 Should Reduce give all cases when $\sqrt{x y} = \sqrt x \sqrt y$?

Is my understanding of Reduce wrong or should Reduce have given the other two cases?

V 12 on windows.

1 Answer

As Coolwater says in his comment, using the domain specification Reals means that all function values are constrained to be real. Clearly Sqrt[x] is not real when $ x<0$ . Instead, constrain x and y to be real using Element:

Reduce[Sqrt[x y] - Sqrt[x] Sqrt[y] == 0 && (x|y) ∈ Reals, {x,y}] //Simplify

(y ∈ Reals && x > 0) || x == 0 || (x <= 0 && y >= 0)

Let’s block ads! (Why?)

Recent Questions – Mathematica Stack Exchange

$\sqrt{x, Cases, give, reduce, Should, \sqrt
  • Recent Posts

    • The Neanderthals
    • What the swarm of new Azure announcements mean
    • Importance of Integrating SharePoint and Dynamics 365 in an SMB
    • InfoWars Surrenders
    • Invest Your Time in the Right Skills to Become a Data Scientist in 2021
  • Categories

  • Archives

    • March 2021
    • February 2021
    • January 2021
    • December 2020
    • November 2020
    • October 2020
    • September 2020
    • August 2020
    • July 2020
    • June 2020
    • May 2020
    • April 2020
    • March 2020
    • February 2020
    • January 2020
    • December 2019
    • November 2019
    • October 2019
    • September 2019
    • August 2019
    • July 2019
    • June 2019
    • May 2019
    • April 2019
    • March 2019
    • February 2019
    • January 2019
    • December 2018
    • November 2018
    • October 2018
    • September 2018
    • August 2018
    • July 2018
    • June 2018
    • May 2018
    • April 2018
    • March 2018
    • February 2018
    • January 2018
    • December 2017
    • November 2017
    • October 2017
    • September 2017
    • August 2017
    • July 2017
    • June 2017
    • May 2017
    • April 2017
    • March 2017
    • February 2017
    • January 2017
    • December 2016
    • November 2016
    • October 2016
    • September 2016
    • August 2016
    • July 2016
    • June 2016
    • May 2016
    • April 2016
    • March 2016
    • February 2016
    • January 2016
    • December 2015
    • November 2015
    • October 2015
    • September 2015
    • August 2015
    • July 2015
    • June 2015
    • May 2015
    • April 2015
    • March 2015
    • February 2015
    • January 2015
    • December 2014
    • November 2014
© 2021 Business Intelligence Info
Power BI Training | G Com Solutions Limited