Consider the axiomatic system and theorem below:



• Axiom 1: If there is a pair of points, then they are on a line together.

• Axiom 2: If there is a line, then there must be at least two points on it.

• Axiom 3: There exist at least two distinct points.

• Axiom 4: If there is a line, then not all the points can be on it.


Theorem 1: Each point is on at least two distinct lines.



A. List all undefined terms involved in the given axiomatic system, including all elements and relations.


B. Explain how the axioms require that the system has three distinct points.


Note: The use of models may be helpful in developing this explanation.


C. Prove theorem 1 for three points, using only the provided axioms.


Note: You do not need to use all four axioms.