Interpret \"Q is between P and R\" by d_t(P,Q)+d_1(Q,R)=d_T(P,R). Color the set

Question

Interpret "Q is between P and R" by d_t(P,Q)+d_1(Q,R)=d_T(P,R). Color the set of all points that are between P(3,1) and R(4,3). Consider the axiomatic system with undefined terms point, line, and on and the following axioms i) There is a line and a point that is not on the line. ii) Every two distinct points have a unique line on them both. iii) Every two distinct lines have at least one point on them both. iv) Every line has at least three points on it. a. The following is a model for the system in which a point is represented by a dot and a line by a line segment, except that the two segments together connecting points F, E, and D is interpreted as one line. This shows that the axiomatic system is (consistent, independent, or complete). b. Show that this system has at least 7 points. c. Show that Axiom iv) is independent of the rest of the axioms. d. Show that Axiom iii) is independent of the rest of the axioms 4. If an axiomatic system is independent, must it be complete? Explain your answer.

Explanation / Answer

a)The model shows that the axiomatic system is consistent as the model shows all axioms are true

i.e. i) as per 1st axiom every line has a point that is not on it eg. line Ab and point C

ii) as per 2nd axiom every two pints has a unique line on them both, eg. 2 point F, C have line FC on them both

iii)as per 3rd axiom every 2 lines have one point on them, eg. line AB and line BC have point B on both of them

iv) as 4th axiom every line has 3 points on it eg. line AC has 3 point A, E, and C on it

b) The model shows that there are 7 distinct lines in it

Axiom (iii) say Every 2 distinct lines have atleast one point on both of them

which means there has to be atleast 7 points to be in common for every 2 lines.

c) axiom (iv) is that every line has atleast 3 points on it

the 1st axiom is about the line and a point not on line so it has no relation with axiom (iv)

the 2nd axiom is about 2 distinct point on 1 line but it doesn't state anything about more than 2 points. So, it has no relation with axiom (iv)

the 3rd axiom is about 2 distinct line and a point on them both so this is also not related to axiom (iv)

as we see it is not possible to prove that axiom (iv) is true as a consequence of rest axioms it is independent of rest axioms

d) As like (c) we can prove that axiom (iii) is independent of rest axioms as non of them can be proved to be true as consequence of other.

e)

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