True or False in Geometry

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taisuke
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True or False in Geometry

Postby taisuke » Fri Sep 07, 2012 3:51 pm UTC

Ok, I can't believe this became an issue and started a discussion with my superior:

True or False

Two lines that have no common points are parallel.

My answer is false, having skew lines in mind.

He said it's true because I should just stick to the statement, and that parallel lines do not have common points. He was questioning why I was bringing up skew lines.

Now, here comes a friend telling me that "it depends" and that "sometimes true" does not automatically makes a statement false.

***

While thinking of an analogy to show my point to my boss, I thought of:

An animal that has wings is a bird.

with "false" as the answer in mind since bats and many other animals have wings as well.

Now the same friend argued its contingently true and that answering false is wrong. And that there's a problem with the test construction.

So I'm left confused because I was certain that having just one counterexample is enough to declare a statement to be false.

It has come to this, I trust in the credibility of the members here in xkcd since I got a lot of homework helps here before.

So please, do enlighten me. Am I really wrong for answering false? Or is there really a mistake on having such item in a True or False test.

Thanks.

Twistar
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Re: True or False in Geometry

Postby Twistar » Fri Sep 07, 2012 4:05 pm UTC

From what I can tell you are correct. Writing it out a little bit more formal logically:

A: IF an animal has wings THEN it is a bird
B: Bats are animals with wings AND bats are not birds
Therefore A is false

The same argument works for your skew vs. parallel lines argument

I think the mistake here is mistaking the converse. both of the converses of these statements are true.

If two lines are parallel then they have no common points
all birds have wings.

Just because A implies B doesn't mean B implies A.

formal logic terms like IF sometimes have fuzzy meanings in regular English, but are well defined when we're talking about math.

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Quizatzhaderac
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Re: True or False in Geometry

Postby Quizatzhaderac » Fri Sep 07, 2012 4:57 pm UTC

Two (straight) lines (not line segments) on the same plane either intersect or are parallel.
To show this we can define the y axis to be line B.
If (x,y) and (x+d,y+1) are points on line A: if d = 0 zero then A is parallel to B.
If (x xor d are positive) then for some value of n: x+nd=0; therefore A must have some point (0, y + n), which is a point on the x-axis (and B).
If (x is positive iff d is positive) then for some value f n: x-nd=0; therefore A must have some point (0, y - n), which is on the x-axis (and B).

However outside of a formal math environment "line" very often means line segment, which is probably what you were thinking.

But no, your boss didn't really prove that two lines with no common points are parallel.
Last edited by Quizatzhaderac on Mon Aug 18, 2014 8:43 pm UTC, edited 1 time in total.
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Snark
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Re: True or False in Geometry

Postby Snark » Fri Sep 07, 2012 5:08 pm UTC

taisuke wrote:Two lines that have no common points are parallel.

My answer is false, having skew lines in mind.
Two lines (not segments) in a plane with no common points are always parallel. Thus the answer is true.

Two lines (not segments) in (3D space or a higher) with no common points are parallel or skew. Thus the answer is false.

Depends on information not explicitly given.
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Re: True or False in Geometry

Postby skullturf » Fri Sep 07, 2012 5:21 pm UTC

Yeah, all the stuff about logic in the OP sounded right.

But if the supervisor was talking only about a two-dimensional space, then there's no such thing as skew lines.

fishfry
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Re: True or False in Geometry

Postby fishfry » Fri Sep 07, 2012 5:25 pm UTC

This is more a question of philosophy or semantics or rhetoric.

If I ask you: True or false, two lines that never meet are called parallel; then you are perfectly well justified in saying, "Yes of course that's true. By definition, two lines that never meet are parallel."

And then I say, "Ha ha fooled you, I'm thinking about three-dimensional space and skew lines ha ha you're wrong!"

Well, that's a "gotcha" question, not a matter of logic.

If you're arguing with your boss over this, I recommend saying to them: "Oh, I see you're right! Got me!!" and leave it alone. What's the point of pushing your boss's buttons over something silly?

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el matematico
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Re: True or False in Geometry

Postby el matematico » Fri Sep 07, 2012 5:50 pm UTC

You also need to agree whether you agree or not on Euclid's fifth postulate. On hyperbolic geometry you have parallel and ultraparallel lines. Also, in any geometry where parallel lines are defined, lines are parallel to themselves, making the statement "parallel lines don't touch" false in general.
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z4lis
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Re: True or False in Geometry

Postby z4lis » Fri Sep 07, 2012 8:04 pm UTC

There's no such thing as a "sometimes true" mathematical statement. If there's a single counterexample, the statement is false. Period. The statement "If two lines do not meet, they are parallel." is false with any reasonable interpretation of "line", and you're completely justified in bringing up the example of skew lines, as lines in 3-dimensional Euclidean space are just as much of lines as lines in 2-dimensional Euclidean space. I don't understand what the problem is.
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dudiobugtron
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Re: True or False in Geometry

Postby dudiobugtron » Fri Sep 07, 2012 10:43 pm UTC

z4lis wrote:There's no such thing as a "sometimes true" mathematical statement. If there's a single counterexample, the statement is false. Period.

Actually, that statement is false. Here's a counter example for you! ;)

"A implies B" is a mathematical statement which is true whenever B is true or A is false. It's not true when A is true and B is false though. So I think it's fair to say that it is "sometimes true".
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taisuke
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Re: True or False in Geometry

Postby taisuke » Sat Sep 08, 2012 9:27 am UTC

fishfry wrote:If you're arguing with your boss over this, I recommend saying to them: "Oh, I see you're right! Got me!!" and leave it alone. What's the point of pushing your boss's buttons over something silly?


I'm a Math teacher and I was shocked that my boss (also a Math teacher) was saying that my answer "False" was wrong. Skew lines were also taught to the students so they were not confined to 2 dimensional idea of lines. I'm 100% sure that I'm right with my answer but the long discussion fed me up. I started having doubts so I asked my queries here.

The statement about animals with wings was contested by a friend who's also a math teacher. He bombarded me with reasons using lessons from Logic that admittedly, I've mostly forgotten.

Anyway, thanks to everyone who replied.

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PM 2Ring
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Re: True or False in Geometry

Postby PM 2Ring » Sat Sep 08, 2012 3:42 pm UTC

This is a #169 situation.
Black Hat Guy wrote:Communicating badly then acting smug when you're misunderstood is not cleverness.


I recommend printing out that comic and putting it in your boss's office. :twisted:

fishfry
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Re: True or False in Geometry

Postby fishfry » Sat Sep 08, 2012 5:45 pm UTC

taisuke wrote:
fishfry wrote:If you're arguing with your boss over this, I recommend saying to them: "Oh, I see you're right! Got me!!" and leave it alone. What's the point of pushing your boss's buttons over something silly?


I'm a Math teacher and I was shocked that my boss (also a Math teacher) was saying that my answer "False" was wrong. Skew lines were also taught to the students so they were not confined to 2 dimensional idea of lines. I'm 100% sure that I'm right with my answer but the long discussion fed me up. I started having doubts so I asked my queries here.

The statement about animals with wings was contested by a friend who's also a math teacher. He bombarded me with reasons using lessons from Logic that admittedly, I've mostly forgotten.

Anyway, thanks to everyone who replied.


Ah! "Skew lines were taught."

Every problem contains implicit context. If I'm in Euclidean geometry class and someone asks about lines that never meet being parallel, that's one thing. If I'm in 3-D geometry class and the teacher talked about skew lines in class yesterday, that's a completely different situation.

If I ask, does -1 have a square root, and we're in high school algebra class, the answer is no. If we're in complex variables class, the answer is yes. If we're in abstract algebra class, the answer is sometimes; for example in the integers mod 5, 2^2 = -1. But in the integers mod 7, there's no such solution.

The context of a problem always matters.

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Re: True or False in Geometry

Postby mathmannix » Fri Sep 21, 2012 5:49 pm UTC

Twistar wrote:all birds have wings.


I know I'm being a picker of nits here, and also that this was not the original point of this thread, but is this actually true? Yes, I know that even flightless birds, from ostriches to kiwis to penguins to the late dodo, have winglike appendages. They have the skeletal structures that are equivalent to a human's arm, or a dog or hog or frog's front legs. But, isn't the definition of a "wing" something that provides flight through a fluid medium (usually air)?

I am willing to grant that the wings of a penguin propel it through water much as the wings of a falcon propel it through air. However, I argue that the kiwi, who has winglike vestigial appendages that are kept hidden beneath its furlike feathers, does not have wings. I think I could argue that ostriches don't really have wings either, but the kiwi is my best example. All flying vertebrates (not merely gliding ones) have wings - I think that limits the statement to flying birds, bats, and extinct pterosaurs.
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fishfry
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Re: True or False in Geometry

Postby fishfry » Fri Sep 21, 2012 9:44 pm UTC

mathmannix wrote:
Twistar wrote:all birds have wings.


I know I'm being a picker of nits here, and also that this was not the original point of this thread, but is this actually true? Yes, I know that even flightless birds, from ostriches to kiwis to penguins to the late dodo, have winglike appendages. They have the skeletal structures that are equivalent to a human's arm, or a dog or hog or frog's front legs. But, isn't the definition of a "wing" something that provides flight through a fluid medium (usually air)?



I understand your point. I'd like to offer a possible counterexample to your thesis.

You are saying, if I understand correctly, that an artifact that fails to serve its intended purpose is by definition not an instance of its general class; by virtue of its failing to operated as intended.

In other words if I try to build a car, if I get a frame and a chassis and and engine and a transmission and I get everything all put together, and I turn the key and the car doesn't start ... then by your thesis, this is not a car.

But I would argue that it's very much a car. Just one that doesn't work. One that, through faulty construction or design or human error, may never work. But it's still a car. In my opinion, anyway.

I think a vestigial wing is nature's idea of a wing that doesn't quite work yet. A little more tinkering should do it. There, the non-car started. Thereby becoming a car! But I say it was a wing all along, and a car all along.

Did I understand you right? Did that make sense?

CatOfGrey
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Re: True or False in Geometry

Postby CatOfGrey » Sat Sep 22, 2012 6:00 am UTC

taisuke wrote:Ok, I can't believe this became an issue and started a discussion with my superior:

True or False

Two lines that have no common points are parallel.

My answer is false, having skew lines in mind.

He said it's true because I should just stick to the statement, and that parallel lines do not have common points. He was questioning why I was bringing up skew lines.

Now, here comes a friend telling me that "it depends" and that "sometimes true" does not automatically makes a statement false.


What you are stating here is a definition, and definitions are not merely "If p, then q" but rather "If and only if p, then q."

So, to break this down, and assuming Euclid, parallel lines are two lines that have no common points. However, two lines that have no common points aren't necessary parallel.

An alternate is perhaps to say that "If two lines that have no common points are parallel, then your space is Euclidean and 2-D."

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Re: True or False in Geometry

Postby Mindworm » Mon Sep 24, 2012 1:37 pm UTC

CatOfGrey wrote:An alternate is perhaps to say that "If two lines that have no common points are parallel, then your space is Euclidean and 2-D."


That's perhaps too strong a statement. I'm pretty sure there are other geometries which make that statement true, if only by virtue of having no lines that have no common point. Such as the one point compactification of the complex numbers.
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Re: True or False in Geometry

Postby MartianInvader » Tue Sep 25, 2012 9:37 pm UTC

dudiobugtron wrote:
z4lis wrote:There's no such thing as a "sometimes true" mathematical statement. If there's a single counterexample, the statement is false. Period.

Actually, that statement is false. Here's a counter example for you! ;)

"A implies B" is a mathematical statement which is true whenever B is true or A is false. It's not true when A is true and B is false though. So I think it's fair to say that it is "sometimes true".


That's not a mathematical statement. That's a... what's the term... formula, I think? The point is it has unbound variables that need to be substituted for before the statement has meaning on its own. None of the statements the OP discussed had this sort of thing.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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Re: True or False in Geometry

Postby dudiobugtron » Tue Sep 25, 2012 11:37 pm UTC

MartianInvader wrote:
dudiobugtron wrote:
z4lis wrote:There's no such thing as a "sometimes true" mathematical statement. If there's a single counterexample, the statement is false. Period.

Actually, that statement is false. Here's a counter example for you! ;)

"A implies B" is a mathematical statement which is true whenever B is true or A is false. It's not true when A is true and B is false though. So I think it's fair to say that it is "sometimes true".


That's not a mathematical statement. That's a... what's the term... formula, I think? The point is it has unbound variables that need to be substituted for before the statement has meaning on its own. None of the statements the OP discussed had this sort of thing.


I thought that was exactly what the OP was discussing; whether "Two lines that have no common points are parallel" means:
a) "In 2D Euclidean space, two lines that have no common points are parallel" (Boss's interpretation, no unbounded variables),
b) "For all possible geometries, two lines that have no common points are parallel" (OP's and z4lis's interpretation),
or:
c) "In the geometry you're currently interested in, two lines that have no common points are parallel." (friend's interpretation, includes an 'unbounded variable').

If it's c, then it is true for some geometries and not for others.
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