Question

Consider two postulates given below:
(1) Given any two distinct points A and B. there exists a third point which is in
between A and B.
(2) There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent?
Do they follow from Euclid's postulatcs? Explain.​

Answer 1

Correct question:-

Consider two ‘postulates’ given below:Given any two distinct points A and B, there exists a third point C which is in between A and B.There exist at least three points that are not on the same line.

Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.

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$\huge\star{\blue{\underline{\underline{\mathfrak{Explanation:}}}}}$

Yes, these postulates contain undefined terms. Undefined terms in the postulates are:

• $\huge{\green\leadsto}$ There are many points that lie in a plane. But, in the postulates given here, the position of the point C is not given, as of whether it lies on the line segment joining AB or not.

• $\huge{\purple\leadsto}$ On top of that, there is no information about whether the points are in same plane or not.

Yes, these postulates are consistent when we deal with these two situation:

• $\huge{\pink\leadsto}$ Point C is lying on the line segment AB in between A and B.

• $\huge{\blue\leadsto$ Point C does not lie on the line segment AB.

$\huge{\red\leadsto}$ No, they don’t follow from Euclid’s postulates. They follow the axioms.

Answer 2

Yes, these postulates contain undefined terms. Undefined terms in the postulates are:

– There are many points that lie in a plane. But, in the postulates given here, the position of the point C is not given, as of whether it lies on the line segment joining AB or not.

– On top of that, there is no information about whether the points are in same plane or not.

Yes, these postulates are consistent when we deal with these two situations:

– Point C is lying on the line segment AB in between A and B.

– Point C does not lie on the line segment AB.

No, they don’t follow from Euclid’s postulates. They follow the axioms.