Proper notation for writing all points in the 1st and 3rd cartesian quadrant?

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Homework Help Overview

The original poster seeks proper notation for expressing the set of all ordered pairs of real numbers located in the first and third quadrants of the Cartesian plane.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster considers a notation involving inequalities and questions whether the use of a plus operation might inadvertently include points from the second and fourth quadrants. Some participants suggest using "union" instead of "plus" for set notation. Others point out the importance of distinguishing between points in quadrants and those on the axes, raising questions about the correct use of inequalities.

Discussion Status

Participants are actively discussing the appropriate notation and clarifying the definitions of the quadrants. There is acknowledgment of potential errors in the original poster's understanding, and suggestions for alternative expressions are being explored.

Contextual Notes

There is a note regarding the distinction between points in quadrants versus those on the axes, as well as a discussion about the implications of including or excluding the axes in the notation.

PhillipKP
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Homework Statement



What is the proper notation for writing the set of all ordered pairs of real numbers that are in quadrant 1 and 3 of the real plane?

Homework Equations





The Attempt at a Solution



I was thinking something like

[tex]$\left\{ \left(x_{1},x_{2}\right)\in\mathbb{R}\,:\, x_{1},x_{2}\geq0\right\} +\left\{ \left(x_{1},x_{2}\right)\in\mathbb{R}\,:\, x_{1},x_{2}\leq0\right\} $[/tex]

Is this fine or will including the + operation mean I am including points in the 2nd and 4th quadrant?
 
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Since you're working with sets, you would want "union" rather than "plus."
 
Note: points in quadrants are not on axes and vice-versa, so you need to be careful with the use of >= and >.

It is also possible to describe the 1st and 3rd quadrants as

[tex]\{(x,y) \in \mathbb{R}^2:xy<0\}[/tex]

If you wish to include the axes, change the inequality to xy <= 0.

--Elucidus
 
Thanks for the quick replies.
 
ERROR!

1st and 3rd should be xy > 0. The coordinates must be of the same sign.

Sorry, somehow I was thinking 2nd and 4th (sheesh, I feel silly).

--Elucidus
 
No worries I figured it was a typo :)

Thank you for your time.
 

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