Algebra
Homework 7 (419):
1: Let X be a set of infinite cardinality κ, and call a set Y ⊂ X “small” if
there is a decomposition of X into subsets of cardinality κ each of which
intersects Y in at most one point. Then X is the union of two of its “small”
subsets.
2: Show that: If X is of cardinality κ ≥ ℵ0, then the following sets are of cardinality κ:
a) set of finite sequences of elements of X,
b) set of those functions that map a finite subset of X into X.
3: Show that: An infinite cardinal is regular if and only if κ is not the sum of fewer than
κ cardinals each of which is less than κ
4: Show that: A successor cardinal is regular.
5: Which are the smallest three singular (i.e., not regular) infinite cardinals?
6: Show that: If α is the cofinality of an ordered set, then α is a regular cardinal.
7: Show that: (Each part is one point)
(a) Every element of an N-set is an N-set.
(b) If x is an N-set, then y = x ∪ {x} is an N-set, and if z is an N-set
containing x, then y ⊂ z.
(c) If x is an N-set, y ∈ x, then y is an initial segment of x.
(d) If x is an N-set and Y ⊂ x is one of its initial segments, then Y is an
N-set, and either Y = x or Y ∈ x.
(e) If x, y are N-sets, then x = y or x ∈ y or y ∈ x.
(f) For N-sets x, y define x < y if x ∈ y. Then this is irreflexive, transitive
and trichotomous. Furthermore, if B is a nonempty set of N-sets, then
there is a smallest element of B with respect to < (“well order”).
(g) If x, y are different N-sets, then they are not similar.
(h) Every well-ordered set is similar to a unique N-set.
8: Show that: There is no infinite decreasing sequence of ordinals.
9: Show that: Arbitrary infinite sequence of ordinals includes an infinite nondecreasing
subsequence.
10: Show that: Addition among ordinals is monotonic in both arguments, and strictly
monotonic in the second argument. The same is true of multiplication
provided the first factor is nonzero.
