Please have a look at the Hasse diagram -

How come the upper bounds of {a,b,c} are e,f,j and h and lower bound is a

Why doesn't it include d and g also in its upper bound?

can anyone please clarify this?

Printable View

- June 15th 2008, 11:45 AMrobocop_911Hasse Diagram - lower upper bound
Please have a look at the Hasse diagram -

How come the upper bounds of {a,b,c} are e,f,j and h and lower bound is a

Why doesn't it include d and g also in its upper bound?

can anyone please clarify this? - June 15th 2008, 12:03 PMPlato
It does not include d because the pair (c,d) is not in the diagram.

There is no g in the diagram. - June 15th 2008, 12:09 PMrobocop_911
- June 15th 2008, 12:19 PMPlato
As I said before, the pairs (c,d) & (c,g) are not in the graph.

The elements g & d are not comparable to c in the poset.

**There is neither an “upward path” from c to d nor one from c to g.**

- June 15th 2008, 12:30 PMrobocop_911
- June 15th 2008, 12:33 PMPlato
Are elements g & j are comparable?

- June 15th 2008, 12:42 PMrobocop_911
- June 15th 2008, 12:46 PMPlato
- June 15th 2008, 12:54 PMrobocop_911
- June 15th 2008, 12:59 PMPlato
- June 15th 2008, 01:04 PMrobocop_911
- June 15th 2008, 01:48 PMrobocop_911
Okay, here are my questions...

1) {b,d,g} upper bounds - g and h

my question - why not only "h" why include g in it?

2) {b,d,g} least upper bound - g

my question - How come it is "g" why not "h" since h>g

3) {b,d,g} lower bounds - a and b

my question - why not only "a" why include b in it?

4) {b,d,g} greatest lower bound - b

my question - why not "a" since a<b

5) {a,c,d,f} upper bounds - f,h and j

my question - why include "f" in upper bound? Also, there is no "upward" connection between a and d. - June 15th 2008, 02:12 PMPlato
1) {b,d,g} upper bounds - g and h

my question - why not only "h" why include g in it?

Because**both**succeed every term in {b,d,g}. Don’t forget that partial orders are reflexive, therefore (g,g).

2) {b,d,g} least upper bound - g

my question - How come it is "g" why not "h" since h>g

Because there is only one LUB for any given set and g is the first or least.

3) {b,d,g} lower bounds - a and b

my question - why not only "a" why include b in it?

See 1). It is the same reason: b proceeds each element in the set.

4) {b,d,g} greatest lower bound – b my question - why not "a" since a<b

See 2). It is the same reason: b is the greatest or last upper bound.

5) {a,c,d,f} upper bounds - f,h and j

my question - why include "f" in upper bound? Also, there is no "upward" connection between a and d.

"a-b-d is an upward" connection between a and d.

There is an "upward" connection between any term in {a,c,d,f} to f [recall (f,f)]..

- June 15th 2008, 02:59 PMrobocop_911
1) {b,d,g} upper bounds - g and h

my question - why not only "h" why include g in it?

Because**both**succeed every term in {b,d,g}. Don’t forget that partial orders are reflexive, therefore (g,g).

**Then how come upper bounds of {a,b,c} is e,f,j and h and not something like c,e,f,j,h? (c,c) (b,b) that's also reflexive!**

2) {b,d,g} least upper bound - g

my question - How come it is "g" why not "h" since h>g

Because there is only one LUB for any given set and g is the first or least.

**So least upper bound should be from the set given only right? In this case it is {b,d,g} so here it is "b". Is it also the greatest lower bound?**

- June 15th 2008, 03:15 PMPlato
I wish you would learn the definitions and how to use them.

In any case, good luck and good night.