Determine, for each of the following sets, whether or not it is countable justif

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

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