Question

Refer the axioms below

Ques

For each part of the five axioms, decide whether it is true or false for a cylinder. Explain how you know that it is true and/or exactly when and how it fails. Make sure that you consider each possible type of line and every possible configuration of points. Write down the best possible variation of the axiom which is true. Draw pictures where appropriate.
Note: To test the plane separation axiom, try actually cutting the cylinder

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An axiom means "A proposition that commends itself to general acceptance; a well established or universally-conceded principle..." (OED2). Often, one assumes the following statements are true. However, they are not true on every space. Therefore, we will check if each statement is true on each space.

1. The "Incidence Axiom":
   1. There is at least one straight line between two (distinct) points.
   2. There is at most one straight line between two (distinct) points.
2. The "Ruler Axiom":
   1. You can travel any distance along a straight line in either direction.
   2. If you travel along any straight line, you will never go over the same point more than once.
3. The "Plane Separation Axiom":     The complement of a straight line (what you have left when you cut along it) consists of two pieces so that:
   1. If two points lie in difference pieces, the line segment between them crosses the line.
   2. If two points lie in the same piece, the line segment between them does not cross the line
4. The "Protractor Axiom":
   1. There is at least one straight line through any point in any direction.
   2. There is at most one straight line through any point in any direction.
5. The "Mirror Axiom":      There is a global reflection through every straight line.

Answer 1