Medium
Consider two 'postulates' given below:
$(i)$ Given any two distinct points $A$ and $B$,there exists a third point $C$ which is in between $A$ and $B$.
$(ii)$ 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.
$(i)$ Given any two distinct points $A$ and $B$,there exists a third point $C$ which is in between $A$ and $B$.
$(ii)$ 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.
(N/A) Yes,these postulates contain undefined terms such as: 'Point' and 'Line'.
These postulates are consistent because they deal with two different situations:
$(i)$ states that given two points $A$ and $B$,there is a point $C$ lying on the line segment between them.
$(ii)$ states that,given $A$ and $B$,you can take a point $C$ that does not lie on the line passing through $A$ and $B$.
No,these postulates do not follow from Euclid's postulates. However,they follow from the axiom: 'Given two distinct points,there is a unique line that passes through them.'
These postulates are consistent because they deal with two different situations:
$(i)$ states that given two points $A$ and $B$,there is a point $C$ lying on the line segment between them.
$(ii)$ states that,given $A$ and $B$,you can take a point $C$ that does not lie on the line passing through $A$ and $B$.
No,these postulates do not follow from Euclid's postulates. However,they follow from the axiom: 'Given two distinct points,there is a unique line that passes through them.'
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