Are the members of (x,y) if and only if (y = x + 3) & (y = x – 3) an empty set?

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Discussion Overview

The discussion revolves around the question of whether the set of pairs (x, y) defined by the conditions (y = x + 3) and (y = x - 3) is an empty set. The scope includes mathematical reasoning and exploration of the implications of these conditions.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions if the set of integers (x, y) can be defined under the conditions given, suggesting it may be an empty set.
  • Another participant points out a contradiction in the initial definition, noting the inconsistency between referring to integers and real numbers.
  • A participant clarifies that the conditions imply that the two equations do not intersect, leading to the conclusion that the set is empty.
  • One reply proposes a method to check for solutions by equating the two expressions, leading to a contradiction, which supports the claim of an empty set.
  • A later reply introduces a different perspective by referencing modular arithmetic, suggesting that the identity could hold under specific conditions in a modular set.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the conditions and the nature of the set. While some argue that the set is empty based on the contradiction found, others introduce alternative contexts that challenge the initial assumptions.

Contextual Notes

The discussion includes unresolved assumptions about the definitions of the sets involved and the implications of the equations. The introduction of modular arithmetic raises further questions about the conditions under which the identity might hold.

Bob4040
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Is the set of all integers where <x,y> \in R if and only if (y = x + 3) & (y = x – 3) an empty set?
 
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Hey Bob4040 and welcome to the forums.

Your definition is a little vague. You mention integers but then you say x and y are real numbers which is a bit of a contradiction.

Are you trying to ask whether x and y are real numbers if the relation holds?
 
I do not mean all real numbers. R is the set. x and y are members of the set on the condition that (y = x + 3) & (y = x – 3). These two lines never intersect, so I think that means it is an empty set?
 
Ohh I see what you mean now.

Well the easiest way to check is to set them equal to one another and see if you get a solution or a contradiction.

y = x + 3, y = x - 3 implies x + 3 = x - 3 which implies 3 = -3 which is a contradiction, so you have proven that no element exists satisfying the relationship so you have the empty set.
 
Thank you for the help!
 
I know this is not what you meant/intended , Bob4040, but, strictly speaking your

identity would hold if x were in the set called Z_3, where 3=-3 :

http://www.wolframalpha.com/input/?i=Z+mod+3

Then,for all x, x+3=x-3 .
 

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