Sheet 6 Question 1 (how can we prove transitivity for R1)

Question

I'm convinced for the other orders. But unable to select appropriate x,y,z that will hold for transivity. What do the arrows between the numerals imply exactly?

Edit: Why does the natural numbers set include 0 here?

Edit 2: The lecture slides makes mention of bidirected sets which include lower and upper bounds in contrast the exercise only asks about directed sets, is there a difference between the two?

Comments

Why is zero included in the natural numbers? That's a matter of definition. One way to define Natural Numbers is as the set of possible sizes of finite sets. If you intersects two disjoint sets, the intersection is the empty set. It has the size zero. Thus, this definition includes the number zero. However, for the order properties, it does not make a difference whether we call the smallest element 0 or 1.

---

There is often the question whether 0 is a natural number or not. At the end, it is a matter of definition, of course. Considering the person who gave the first axioms to define what natural numbers are, we get the clear answer that 0 is a natural number: https://en.wikipedia.org/wiki/Peano_axioms

---

About "bidirected sets": At first, we defined partial orders, and later we considered special orders like total orders, directed sets, and (complete) lattices. These special orders are not the same, they have their own definitions which distinguishes them from the others. In particular, the exercise should show you different partial orders that have the one or the other property.

Answer 1

The figures show Hasse diagrams and an arrow represents the ordering relation represented by that Hasse diagram. Note however, that a Hasse diagram does not show the entire ordering relation but just the core which does not contain transitive pairs, i.e., (x,z) is not contained if there is a y with (x,y) and (y,z).

Comments

Maybe, our phrasing about directed and bidirected sets diverges between lecture notes and online exercises. (See edit 2 of the question)