Problems(1)
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≤m′ and n≤n′. An example is (5,−10)<(8,−2). Now let O be a completely ordered subset of S, in other words if (a,b)∈O and (c,d)∈O, then (a,b)<(c,d) or (c,d)<(a,b). Also let O′ denote the collection of all such completely ordered sets.(a) Determine whether and arbitrary O∈O′ is finite.
(b) Determine whether the carnality ∣O∣ of O is bounded for O∈O′.
(c) Determine whether ∣O∣ can be countable infinite for any O∈O′. ordered setcardinality