Affine Independence Proof

Affine Axiom 1: A line lies on at least two points.

Proof: Assume that Axiom 1 is dependent on the other affine axioms. Let  be parallel to . Affine Axiom 4 tells us that there are three non-collinear points, . Affine Axiom 2 tells us that any two distinct points have exactly one line in common. Affine Axiom 3 tells us that any point not on a line l lies on exactly one line not intersecting l. Consider  on . Since , we know Axiom 3 is satisfied. However, notice that the diagram above satisfies Axioms 2,3, and 4, but does not satisfy Affine Axiom 1 which says a line lies on at least two points. Therefore, by contradiction, we have shown that Affine Axiom 1 is independent from the other affine axioms.

Affine Axiom 2: Any two distinct points have exactly one line in common.

Proof: Assume that the affine axioms are dependent on one another. Axiom 4 tells us that there exists a set of three non-collinear points. Consider points  and . Axiom 3 tells us that any point not on a line lie on exactly one line not intersecting that line. That means that there are lines  and through . Axiom 1 tells us that a line lies on at least two points. Point  on the line through  accommodates this axiom. However, notice that there does not exist exactly one line through any two distinct points. Therefore, Axiom 2 is independent from the other three affine axioms.

Affine Axiom 3: Any point P not on a line l lies on exactly one line not intersecting l.

Proof: Assume that the axioms are dependent on one another. Affine Axiom 4 tells us that there are three non-collinear points, and . Axiom 2 tells us that there is exactly one line through any two distinct points. This is seen in  and . These lines also follow Affine Axiom 1 which says that a line lies on at least two points. However, note that there is not a line through  parallel to . Therefore, Affine Axiom 3 is independent of the other axioms.

Affine Axiom 4: There exists a set of three non-collinear points.

Proof: Assume that the affine axioms are dependent on each other. Axiom 3 tells us that there exists a point  not on line  that lies on exactly one line  not intersecting . This follows Affine Axiom 1 because a line lies on at least two points. This also follows Affine Axiom 2 because any two distinct points have exactly one line in common. However, notice that there are four points in this sketch in order to satisfy all three of these axioms. This contradicts Affine Axiom 4 which states that there are three non-collinear points because points all lie on one line, . Therefore, Axiom 4 is independent of the other axioms.