Answer true or false and justify If A is countable and B is

Answer true or false, and justify: If A is countable and B is a finite subset

of A, then A B is countable.

Solution

Suppose that AB=AAB is countable. Then you have a bijection between AB and the set of even natural

numbers. Since B is countable then AB is also countable and you have a bijection between NF A_Band

between F and ABwhere F is a finite subset of N of the same cardinality as AB).   Now, taking the union of

those bijections, you get the bijection between the set A=(AB)(AB)and the set of natural numbers, which is

an absurd because A is countable. Thus AB is countable.