Homework 6 (419)

Homework 6: (419)

1: Show that: There is a family of cardinality bigger than continuum of subsets of R that is ordered with respect to inclusion.

2: Verify: Any countable ordered set is similar to a subset of Q ∩ (0, 1).

3: Any countable densely ordered set without smallest and largest elements

is similar to Q.

4: Any countable densely ordered set is similar to one of the sets Q∩ (0, 1),

Q∩[0, 1), Q∩(0, 1], Q∩[0, 1] (depending if it has a first or last element).

5: There is an uncountable ordered set such that all of its proper initial

segments are similar to Q or to Q ∩ (0, 1].

6: Verify: There is an uncountable ordered set which is similar to each of its uncountable

subsets.

7: What is the cardinal a0 · a1 · · · if the ai’s are positive integers?

8: Prove: (Fundamental theorem of cardinal arithmetic) For every infinite cardinal

κ we have κ2 = κ.

9: Show that: If at least one of κ > 0 and λ > 0 is infinite, then

κ + λ = κλ = max{κ, λ}.

10: Show that: The supremum of any set of cardinals (considered as a set of ordinals) is

again a cardinal.