Exercise from Naive Set Theory by Halmos

[tmbrwlf730]

For those who have read Halmos, in Section 6 Ordered Pairs (page 23 in my book), he gives a non-trivial exercise to find an intrinsic characterization of those sets of subsets of A that correspond to some order in A. I'm curious what that characterization is.

A is suppose to be a quadruple {a, b, c, d} and he gives the order c, b, d, a as an example.
C is a set who's elements are sets that for each particular spot in the ordering, that set's elements are those that occur at or before the spot.

So we can write the order in the example above as {c}, {c, b}, {c, b, d}, {c,b,d,a}
and C = { {a, b, c, d}, {b, c}, {b, c, d}, {c}}.

Is the characterization just that C has the same number elements as A?

 

[micromass]

Is the characterization just that C has the same number elements as A?

That is of course necessary but it doesn't characterize these sets. For example

\{\{a\},\{b\},\{c\},\{d\}\}

also has the same number of elements but doesn't correspond to some order in $A$ (I assume you mean a total order).

Look at the set $C$. Is the set $C$ ordered in some way?