Fifith Postulate in Affine Spaces

[rdabra]

If possible, could anyone criticize the following sequence of sentences ?
1) Two n-dimensional affine subspaces are parallel iff they have the
same associated linear space.
2) Two different (non-equal) parallel affine subspaces are disjoint sets.
3) If two n-dimensional affine subspaces ( 'A' and 'B' ) are parallel to a third n-dimensional subspace then they are parallel to each other. (A || B )
4) Euclid's Fifth Postulate: Based on item 3), if two n-dimensional subspaces ( 'A' and 'B' ) parallel to a third n-dimensional subspace have a point in common, then they are equal (A = B).
5) Based on item 4), every affine space obeys the parallel postulate.

My question is : What item is wrong ?
thanx in advance

 

[pat_connell]

.Item 5 is incorrect. Euclid's Fifth Postulate states that, if two lines are parallel to a third line, then they must remain parallel to each other and never meet. This does not mean that "every affine space obeys the parallel postulate". In fact, affine spaces are not necessarily subject to Euclid's Fifth Postulate, as it only states something about lines.

 

[tacman]

The Fifth Postulate in affine spaces states that two n-dimensional affine subspaces are parallel if and only if they have the same associated linear space. This means that the direction of the subspaces is the same, but they may have different translations or offsets. This postulate is often used in geometry and has been criticized for not being intuitive or easily understood.

One possible criticism of this sequence of sentences is that they do not clearly define what is meant by "parallel" in an affine space. While it is stated that parallel subspaces have the same associated linear space, it is not explicitly defined how this is determined. Additionally, the statement that two different (non-equal) parallel subspaces are disjoint sets may be misleading, as it suggests that parallel subspaces cannot intersect at all, which is not necessarily true.

Another potential issue is with the use of the term "equal" in relation to parallel subspaces. In affine spaces, two subspaces can be considered equal if they have the same direction and the same translation or offset. However, this is not the same as being equal in the traditional sense of mathematics. This may lead to confusion or misinterpretation of the postulate.

Furthermore, the use of Euclid's Fifth Postulate in this context may be seen as problematic. While it may be applicable to certain geometric situations, it is not a universally accepted postulate in the study of affine spaces. There may be other postulates or axioms that could also be used to describe the relationship between parallel subspaces.

In conclusion, while the Fifth Postulate in affine spaces may be a useful tool in certain situations, it is not without its limitations and potential criticisms. It is important to carefully define and clarify the terms used in the postulate to avoid confusion and potential misunderstandings.