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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.

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# Homework Help: Math axioms problem

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