Why Are There Contradictions in Hasse Diagram Examples?

[arnold28]

Ok I don't understand one thing...We ahd this example in school

We have poset < {1,2,5,7,10,14,35,70}, | >
| meas factor, for example a|b means b=ka, where k=integer

and we got this Hasse diagram

        70
       / | \
     10 14 35
     | X  X |
     2   5  7
      \  | /
         1

X means crossed lines

But the hasse diagram definition says that we draw a line between element and the element above it if and only if the lower element is the immediate predecessor of the above one. So why is there a line between 1 and 5? Because there's element 2 in between. Same with 1 and 7, 2 and 10, 7 and 35, etc etc same with almost every element. To me they don't seem to be immediate predecessors/successors with each other

I have google some more examples but they all have the same contradiction between definition and example. So what i don't understand here :(

 

[Hurkyl]

So why is there a line between 1 and 5? Because there's element 2 in between.

Really? 1 divides 2 and 2 divides 5?

 

[arnold28]

i still don't get it :(
for example why is 2 immediate predecessor of 10

 

[Hurkyl]

i still don't get it :(
for example why is 2 immediate predecessor of 10

Well, what could come between them? What numbers both divide 10 and are divisible by 2?

 

[blue_raver22]

As a scientist, it is important to understand that contradictions can arise in any field, including mathematics and specifically Hasse diagrams. The purpose of a Hasse diagram is to represent the partial order of a set, where elements are connected if they are immediate predecessors or successors of each other. However, the example given may not fully adhere to this definition.

One possible explanation for the contradictions in this example is that the elements 2 and 5 are not properly labeled as immediate predecessors or successors. This could be due to a mistake in labeling or a misunderstanding of the concept. It is important to carefully consider and label each element in a Hasse diagram to accurately represent the partial order of the set.

Additionally, it is important to note that Hasse diagrams are not unique and can be drawn in different ways. This means that alternate Hasse diagrams can represent the same partial order, but may look different from each other. It is possible that the example given is just one way of representing the partial order and there may be other ways to draw it that do not have the same contradictions.

In conclusion, it is important to carefully consider the elements and their relationships when drawing a Hasse diagram to avoid contradictions. It is also important to understand that there may be different ways of representing the same partial order and not all examples may perfectly adhere to the definition. As scientists, it is our duty to continuously question and analyze information to gain a deeper understanding of complex concepts.