Math Axioms Problem: Solve Questions & Get Hints

[ynuo]

Hello,

Can you please help me with the questions listed below. I would like to get hints on how I can solve them. I have listed first the axioms and then the questions at the bottom.

Axioms:

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A plane consists of:

-two sets P and L such that P && L = phi
-a subset I of P * L.

Given any plane (P,L,I) we make the following definitions:

(1) The elements of P are called points and those of L
are called lines.

(2) Let x be a point and L be a line; to indicate that
(x,L) ?element of I? we say that the point x is on the line L",
or that the line L goes through the point x."

(3) Points x, y, z are said to be collinear if there
exists a line L which goes through x, y, and z.

(4) A common point of lines L1, L2 is a point x which is
on each of L1, L2.

(5) Two lines L1, L2 are said to be parallel if L1 = L2
or if the lines have no common point.

Definition. An affine plane is a plane (P,L,I) which satisfies the following conditions:

Af1: Given distinct points x, y, there exists a unique line which goes through x and y.

Af2: Given a line L and a point x, there exists a unique line L0 which goes through x and which is parallel to L.

Af3: There exist three points which are not collinear.
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Questions:

1) Prove that for any lines L1, L2, L3 in an any plane,
if L1 is parallel to L2 and L2 is parallel to L3 then L1 is parallel to L3.

2) Let x > 0 and y > 0 be real numbers. Show that the pair x, y is commensurable if and only if x/y is a rational number.

 

[arildno]

A very nice list of questions you have there!

Was it so hard to type it in that you didn't have time to do any thinking on your own as to what the answers might be?

 

[ynuo]

Here is what I have done so far:

For the 2nd I believe I got the answer. But for the first one I think that definition 4 contradicts definition 5 and Af3. Because by definition 5:

L1=L2 and L2=L3.

This means according to definition 4 that every point contained by L1 is contained by L2. Also, every point contained by L2 is contained by L3. This implies that L1=L3.

The second part of definition 5 says that two lines are parallel if they have no common points. Suppose that L1 and L2 have no common points. L2 and L3 have no common points. How can I prove that L1 and L3 have no common points. I haven't studied set theory so I am not sure how to do this.

Thanks.