# Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18 Prove that countable union of countable sets is countable.

Question
Discrete math
Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18 Prove that countable union of countable sets is countable.

2021-02-23
According to the given information, it is required to prove that countable union of countable sets is countable. Let  be the sequence of countable sets. Now define: $$S= \bigcup S_{n}n \in N$$ for all n in N denote T, denote the set of all injections from $$S_{n} \text{to} N$$. since $$S_{n}\ \text{is countable so}\ T_{n}$$ is non empty. so, there exist a sequence $$t_{n} \in T_{n}$$ define, $$f:S \rightarrow N \times N$$ be the mapping defined by $$f(x) = (n, t_{n}(x))$$ where n is the smallest unique natural number such that x in $$S_{n}$$ from well ordering principle such an n exist hence mapping f exist. each t_n is an injection, thus it follows that f is an injection. $$N \times N$$ is countable so, there exist a: $$N \times N \rightarrow N$$ since composite of injections is injection thus, a circ $$f:S \rightarrow N$$ is an injection therefore, S is countable

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