Let S be the set of all intervals [a,b] such that a,b subset of {1, 2, 3, 4, 5} and a<b.?

Question

Let relations R1 and R2 be defined as:

xR1y iff x subset of y

xR2y iff x intersect y not an empty set.

a) determine wether or not R1 is a partial ordering on S an wether or not R2 is a partial ordering on S.

b) if (S,R1) and/or (S,R2) is a poset, then draw its Hasse diagram and decide wether it is a lattice.

Answer 1

R1 should be a partial ordering on S.  R2 won&#039;t be ((1) it&#039;s symmetric and (2) it isn&#039;t transitive).

I think (S, R1) will be a join-semilattice, but it doesn&#039;t seem to have a 0-hat, so it can&#039;t be a lattice.