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

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|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|

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