Transcribed Image Text: Note. If B is a subset of A and A is countable then B is countable. To see

Excerpt
|A| means that there is a one-to-one function from B into A. Suppose that B is a subset of A. Specify a one-to-one function whose domain is B and whose range is a subset of A.By doing ts you will show that |B| < |A|

Transcribed Image Text: Note. If B is a subset of A and A is countable then B is countable. To see ts let
a1, a2 . . . be a sequence whose range is A. Ts means that every agis a member of A
and every member of A is ar for some k . Then define the sequence b1, b2, … in the
following way. Let ji be the first integer j such that a, e B. Given that jr has been
found, let jk+1 be the first integer j > jk such that a; e B. Do ts for k = 1, 2, ….
Let br
%3D
ajr
for k = 1, 2,…. The sequence b1, b2 . .. will have range B.
%3!
Note. According to the definition, when each of A and B is a set, |B| < |A| means that there is a one-to-one function from B into A. Suppose that B is a subset of A. Specify a one-to-one function whose domain is B and whose range is a subset of A.By doing ts you will show that |B| < |A|