striaght lines - false/true

shahar

shahar

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Joined
Jul 19, 2018
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The question is:
1755350009397.png

Given:
Line a intersects line b.
Line b intersects line c.
Conclusion:
Line c intersects line a.
The AI give me that isn't true, but I don't understand the explaining, so I publish it to get a human eye.
How can I prove the conclusion is false?
The Hebrew source:
נתון ישר a חותך את ישר b
ישר b חותך את ישר c.

מסקנה: ישר c חותך את ישר a.
 
You can disprove it with a counterexample.

You are probably imagining something like this:

1755355361468.png​

Here line a intersects line b (at C). and line b intersects line c (at A); and line c intersects line a (at B).

But what about this?

1755355646287.png​

Here line a intersects line b (at C). and line b intersects line c (at A); but line c does not intersect line a (because they are parallel).

The important question is, do you understand why your own reasoning, which presumably led you to believe the claim was [always] true, is invalid?
 
Dr.Peterson said:
You can disprove it with a counterexample.

You are probably imagining something like this:


Here line a intersects line b (at C). and line b intersects line c (at A); and line c intersects line a (at B).

But what about this?


Here line a intersects line b (at C). and line b intersects line c (at A); but line c does not intersect line a (because they are parallel).

The important question is, do you understand why your own reasoning, which presumably led you to believe the claim was [always] true, is invalid?
Click to expand...
Now I see it. There is more examples that not based on parallel lines?
 
shahar said:
There is more examples that are not based on parallel lines?
Click to expand...

Tell me what you think.

Can two lines not intersect, without being parallel?

And I'll repeat my other question: Was there a reason you thought the statement was true (In which case you can learn from this not to trust that kind of reasoning), or were you just thinking it might be true because you couldn't think of a counterexample (in which case you can learn not to trust that kind of thinking!).
 
shahar said:
the counterexample is what I sought.
Click to expand...
I think what you're saying, ultimately, is that you want to know how to decide, when you see a statement like this,
shahar said:
Given:
Line a intersects line b.
Line b intersects line c.
Conclusion:
Line c intersects line a.
Click to expand...
whether it is true. What do you do if you don't immediately see a counterexample, and all the examples you can think of are those in which it is true (like my first picture)?

One good place to start is to ask yourself, what would it look like if the conclusion were false? (If that's unimaginable, then you start thinking about why that is, which might lead to a proof.)

Thinking about two lines not intersecting should immediately suggest (in fact, imply!) two parallel lines. So, rather than start by imagining the givens and picturing the usual case, you can start by imagining two parallel lines a and c, and filling in what line b might be in order to satisfy the givens. You'll find yourself drawing something like my counterexample.
 
Dr.Peterson said:
I think what you're saying, ultimately, is that you want to know how to decide, when you see a statement like this,

whether it is true. What do you do if you don't immediately see a counterexample, and all the examples you can think of are those in which it is true (like my first picture)?

One good place to start is to ask yourself, what would it look like if the conclusion were false? (If that's unimaginable, then you start thinking about why that is, which might lead to a proof.)

Thinking about two lines not intersecting should immediately suggest (in fact, imply!) two parallel lines. So, rather than start by imagining the givens and picturing the usual case, you can start by imagining two parallel lines a and c, and filling in what line b might be in order to satisfy the givens. You'll find yourself drawing something like my counterexample.
Click to expand...
The reason of the purpose to ask the question is I take a long time period break of solving geometric problems, so I try too make a path to get full comprehension of what I learnt and I (how-to-say) forget (and the base is need to get strenghted).
 
"Thinking about two lines not intersecting should immediately suggest (in fact, imply!) two parallel lines."

You could also think about skew lines if another dimension is allowed.
 
This is one problem with AI - you must define the problem exactly. You must include - " No 3-D consideration" to get the answer you are expecting (for this problem).
 
Just to make sure we aren't adding confusion:

The original question was,
shahar said:
Given:
Line a intersects line b.
Line b intersects line c.
Conclusion:
Line c intersects line a.
The AI give me that isn't true, but I don't understand the explaining, so I publish it to get a human eye.
How can I prove the conclusion is false?
Click to expand...
The AI is correct in saying the conclusion does not follow from the givens, whether the context is planar or not. Presumably that is the expected answer.

The easiest proof of that is a counterexample where a and c are parallel (everything in one plane), and that is sufficient. More generally, a and c could be skew lines that both intersect b, though it isn't necessary to think in that direction, and that would not be appropriate if the problem were explicitly in plane geometry.

If we were told what explanation the AI gave, there might be more to discuss.
khansaheb said:
This is one problem with AI - you must define the problem exactly. You must include - " No 3-D consideration" to get the answer you are expecting (for this problem).
Click to expand...
Of course, this is actually the problem with any problem! Whether AI is involved or not, an exact statement is needed. And when we are not told the context, we need to ask. If it actually mattered, hopefully we, or the AI, would have asked.

R.M. said:
"Thinking about two lines not intersecting should immediately suggest (in fact, imply!) two parallel lines."

You could also think about skew lines if another dimension is allowed.
Click to expand...
You're exactly right. Without knowing this, my word "suggest" is appropriate, but "imply" was not. I was assuming the simplest possible context.

Knowing the context would certainly help, though it wouldn't change the answer.
 
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