MathDB

Let

S

be the set of all ordered pairs of integers

(m,n)

satisfying

m>0

and

n<0.

Let

<

be a partial ordering on

S

defined by the statement

(m,n)<(m',n')

if and only if

m\le m'

and

n\le n'.

An example is

(5,-10)<(8,-2).

Now let

O

be a completely ordered subset of

S,

in other words if

(a,b)\in O

and

(c,d) \in O,

then

(a,b)<(c,d)

(c,d)<(a,b).

Also let

O'

denote the collection of all such completely ordered sets.
(a) Determine whether and arbitrary

O\in O'

is finite. (b) Determine whether the carnality

|O|

O

is bounded for

O\in O'.

|O|

can be countable infinite for any

O\in O'.

Problem Statement