Question

a) Prove that the union between two countably infinite sets is a countably infinite set.

b) Would the statement above hold if we instead started with an infinite amount of countably infinite sets?

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Thank you in advance!

Answer 1

a) Let A and B be two countably infinite sets. This implies that there exists bijections . Then, let us look at the function , where

Note that if x is in A, then all of the even numbers are covered and if x is in B\A, then all the odd numebrs will be covered. Hence, . Now, as f and g are bijective, hence h is injective. Thus h is a bijection therefore is a countably infinite set.

b) The result would still hold if we take countable number of countably infinte sets, i.e. the cardinality would be the same as as if you keep doing unions of the sets, the cardinality remains the same due to the two set case in part a).