Finding a counterexample, apparently incorrecct.

[11204]

The question was to find a counterexample for this:
XY = YZ, then Y is the midpoint of Line Segment XZ. While there were many extremely easy counterexamples for this, such as an angle or triangle, I had a very different way of explaining it. I said that so long as the measure of XY was equal to that of YZ, then XZ did not have to be attached or even have Y lie on it. Of course, my teacher told me this was incorrect. This is because Y and X cannot be in two places. I argued that XY could exist in Universe A, YZ could exist in Universe B and XZ In Universe C. I have other ways of backing up what I said, but does anybody have something that will help me see why I'm wrong, and also maybe a way I could have been right.

 

[JeffM]

The question was to find a counterexample for this:
XY = YZ, then Y is the midpoint of Line Segment XZ. While there were many extremely easy counterexamples for this, such as an angle or triangle, I had a very different way of explaining it. I said that so long as the measure of XY was equal to that of YZ, then XZ did not have to be attached or even have Y lie on it. Of course, my teacher told me this was incorrect. This is because Y and X cannot be in two places. I argued that XY could exist in Universe A, YZ could exist in Universe B and XZ In Universe C. I have other ways of backing up what I said, but does anybody have something that will help me see why I'm wrong, and also maybe a way I could have been right.

The convention is that, in the same problem, Y in XY and Y in YZ represent the same point. Any other convention would be highly ambiguous.

Therefore X and Z are connected. You could draw from X to Y and then from Y to Z without lifting your pencil, let alone changing universes.

 

[11204]

I understand that. However, is there any way what I said could be considered correct? Why EXACTLY is what I said incorrect?

 

[JeffM]

I understand that. However, is there any way what I said could be considered correct? Why EXACTLY is what I said incorrect?

You would be correct if your premise that Y is not the same as Y was accepted.

If I say, "A bear may be a vegetable." And you respond "Oh no. A bear is a large omnivorous mammal, and so it is not a vegetable." And I rejoin, "I agree that a bear may be a large omnivorous mammal, but, in my language, a bear may also be what you in your universe call a carrot, and so it may be a vegetable or an animal, no way to say in general." In terms of my language, I am perfectly correct, and you are wrong to say that a bear is not a vegetable.

You are missing the social or super-individual dimension of language, including mathematical language. There must be agreement on what terms mean before any meaningful communication is possible. You are certainly legally within your First Amendment rights to call a bear a vegetable, but everyone else is equally within their rights to regard you as spouting nonsense. In mathematical language, within the context of a given theorem or problem, the convention is that the point Y is a single point, not two or more different points.

 

[Mrspi]

The question was to find a counterexample for this:
XY = YZ, then Y is the midpoint of Line Segment XZ. While there were many extremely easy counterexamples for this, such as an angle or triangle, I had a very different way of explaining it. I said that so long as the measure of XY was equal to that of YZ, then XZ did not have to be attached or even have Y lie on it. Of course, my teacher told me this was incorrect. This is because Y and X cannot be in two places. I argued that XY could exist in Universe A, YZ could exist in Universe B and XZ In Universe C. I have other ways of backing up what I said, but does anybody have something that will help me see why I'm wrong, and also maybe a way I could have been right.

I DON'T agree with your "different universe" example....but you were pretty much on the right track when you said "Y does not have to be on XZ". If X and Z are two distinct (different) points, then they determine exactly one line which contains both of them. That's one of the axioms (or postulates) of Euclidian Geometry. Now, if you pick as Y a point that is equally distant from X and Z (so that XY = YZ) but with Y NOT on XZ, then Y is not going to be the midpoint of XZ. The midpoint of segment XZ, by definition, is the point ON segment XZ for which it is true that XY = YZ. If Y is not on segment XZ, then it won't be the midpoint of XZ even though it might be the same distance away from both X and Z.

 

[JeffM]

Having read Mrspi's correct and clearly expressed comment, I'd like to supplement my comments.

XY and YZ having equal length does not imply that Y lies on line segment XZ. And if Y does not lie on line segment XZ, then clearly it is not the midpoint of line segment XZ. That argument can also be expressed as the possibility that XYZ defines a triangle or an angle not equal to 180 degrees. These are just different ways to express the same thought though Mrspi's seems to me the clearest.

All I was addressing was your proposed distinct argument that somehow the Y in XY can be considered to be referring to a different point from the Y in YZ, perhaps points in two distinct planes or even two distinct universes. That would be inconsistent with the standard conventions of mathematical communication. So that argument, I believe, is not valid because it violates the implicitly agreed conventions. It is changing the rules of the game.

Your answer would be a valid response to the question:

\(\displaystyle If\ the\ length\ of\ line\ segment\ XY_1 = the\ length\ of\ line\ segment\ Y_2Z,\)

\(\displaystyle must\ either\ Y_1\ or\ Y_2\ be\ the\ midpoint\ of\ the\ line\ segment\ that\ contains\ X,\ Z,\ Y_1,\ and\ Y_2?\)

But now it is clear that \(\displaystyle Y_1\ and\ Y_2\) may be different points, neither of which necessarily lies between X and Z even though all four

points lie in the same line.

 

[soroban]

Hello, 112041

Your two-universe argument sounds like amateurish science-fiction.

It can be used to argue any statement, however bizarre.


For example: .\(\displaystyle 1 + 1 \:=\:\varkappa\)


Explanation

Suppose the first "1" is from universe A.
. . It represent one gram of matter.

Suppose the second "2" is from universe B.
. . It represents one gram of anti-matter.

Then the combination could result in the annihilation
. . of both universes, represented by \(\displaystyle \varkappa.\)


The answer to Life, The Universe and Everything could be 42.