Lines, points and contradiction

In summary, the conversation discusses the concept of a line with a point removed and how it can be proven that the line is now two rays instead of still being a line. It also mentions the definition of connectedness in topology and how it relates to this concept.
  • #1
nate808
542
0
this question may sound kind of dumb, so if it is, i apologize, but let say you have a line with a point taken out of it. (<------- -------->) since a point does not have a defined length, how could you prove that the line is now two rays, and not still a line, because it seems a bit contradictory to say that something isn't connected but yet has no distance between it
 
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  • #2
It is very simple, the parametrization (Is this word English?) cannot be a continuous (Is this word Maths?) function, then by definition, it is not a curve.
 
  • #3
Topologically a set is called connected if it can not be devided in two disjoint open subsets. Now consider the set A of a line with one point removed from it (for instance [tex]A = \mathbf{R}- \{ 0 \} [/tex]). We can divide A into two open subsets, namely the two rays (without their starting point). Because they are disjoint, A can not be connected.
 
  • #4
It is incorrect:

[tex]S \subseteq \mathbb{R}^n[/tex] is connected [tex]\leftrightharpoons \exists A, B \subset \mathbb{R}^n : A \cup B = S , A \cap \overline{B} = \varnothing , B \cap \overline{A} = \varnothing[/tex]
 
  • #5
nate808 said:
this question may sound kind of dumb, so if it is, i apologize, but let say you have a line with a point taken out of it. (<------- -------->) since a point does not have a defined length, how could you prove that the line is now two rays, and not still a line, because it seems a bit contradictory to say that something isn't connected but yet has no distance between it


it is unmathematical and meaningless in the english language to say something "has no distance between it"

look at the definitions anyway (whatever those may be for you).
 
  • #6
Maxos said:
It is incorrect:

[tex]S \subseteq \mathbb{R}^n[/tex] is connected [tex]\leftrightharpoons \exists A, B \subset \mathbb{R}^n : A \cup B = S , A \cap \overline{B} = \varnothing , B \cap \overline{A} = \varnothing[/tex]
Well, yes, that is if you do not require A and B to be open in S. Both statements are equivalent though.
(Recall that A open in S ([tex]\subset \mathbb{R}^n[/tex]) means that there exist an open [tex]U \subset \mathbb{R}^n[/tex] such that [tex]U \cap S = A[/tex].)
 

1. What is a line in mathematics?

A line is a straight path that extends infinitely in both directions. It is one of the fundamental objects in geometry and can be described using its slope and y-intercept, or by two points on the line.

2. How are points and lines related?

Points and lines are closely related in mathematics. A line can be thought of as a collection of infinitely many points, and a point can be thought of as the intersection of two or more lines. Points are also used to define the position of a line in space.

3. What is the difference between a line and a line segment?

A line segment is a portion of a line that has two endpoints, while a line extends infinitely in both directions. A line segment has a finite length, while a line has no defined length. Additionally, a line segment can be measured, while a line cannot be measured.

4. Can a line and a point be considered a contradiction?

No, a line and a point cannot be considered a contradiction. A contradiction occurs when a statement or concept is logically opposite or inconsistent. A line and a point are different objects with different definitions, but they are not logically opposite or inconsistent with each other.

5. How are lines and contradictions used in mathematics?

In mathematics, contradictions are used to prove that a statement or assumption is false. Lines are used to represent relationships and patterns in various mathematical concepts, such as geometry and algebra. They are also used in proofs to demonstrate logical connections between different concepts.

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