Consider the axiomatic system and theorem below Axiom 1 For

Consider the axiomatic system and theorem below: Axiom 1: For any two points, there exists a line so that each of the two points is on the line. Axiom 2: There exist at least two points on any line. Axiom 3: There exist at least three distinct points. Axiom 4: Not all points are on the same line. Theorem 1: Each point is on at least two distinct lines. A. List the undefined terms involved in the given axiomatic system. 1. Prove Theorem 1 using the axioms as needed.

Solution

In euclid geometry, the terms LINE and POINT are undefined.

Proof of theorem 1 :   As by axiom one, A line passes from at least two points or on a line there are atleast two points there. Further by axiom four, it is cleared that not all given points lie on the same line, that means few points may lines on two distinct lines. Thus it is cleared that it is not necessary , using Axiom 1 and axiom 4 that it is sufficient for a line to have two given points on it and accordingly other points of the plane may form other distinct points. Thus using these two axioms , it is cleared that each point of the plane is on at least two distinct lines.

Proved