Question
Determine, for each of the following sets, whether or not it is countable justify your answers. (a) The set A of all functions f: {0, 1) rightarrow Z_+. (b) The set of all functions f: (1, ..., n) rightarrow Z_+. (c) The set C = Union_n element Z_+ B_n. (d) The set D of all functions f: Z_+ rightarrow Z_+. (e) The set pound of all functions f: Z_+.rightarrow {0, 1}. (f) The set F of all functions f: Z_+ | [0, 1] that are "eventually zero." [We sat that f is eventually zero if there is a positive integer N such that f (n) = 0 for all n greaterthanorequalto N.] (g) The set G of all functions f: Z_+ rightarrow Z_+ that are eventually 1. (h) The set H of all function f: Z_+ rightarrow Z_+ that are eventually constant. (i) The set I of all two-element subsets of Z_+. (j) The set J of all finite subsets of Z_+.
Explanation / Answer
Solution:
(a) Not Coutable
It is not bijection since domain is limited.
(b) Coutable
It is countably infinte
(c) Countable
It is bijection
(d) Countable
because bijection