Computability and Logic

PROBLEMS 15
the set of all positive integers. (In other words, given an enumeration, which is
to say a function from the set of positive integers onto a set A, show that if A
is not finite, then there is a correspondence, which is to say a one-to-one, total
function, from the set of positive integers onto A.)
1.5 Show that the following sets are equinumerous:
(a) The set of rational numbers with denominator a power of two (when written
in lowest terms), that is, the set of rational numbers ±m/n where n = 1or2
or 4 or 8 or some higher power of 2.
(b) The set of those sets of positive integers that are either finite or cofinite,
where a set S of positive integers is cofinite if the set of all positive integers
n that are not elements of S is finite.
1.6 Show that the set of all finite subsets of an enumerable set is enumerable.
1.7 Let A ={A 1 , A 2 , A 3 ,...} be an enumerable family of sets, and suppose that each
A i for i = 1, 2, 3, and so on, is enumerable. Let ∪A be the union of the family
A, that is, the set whose elements are precisely the elements of the elements of
A.Is∪A enumerable?