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
asked 2021-02-22
Discrete mathematics cardinality using Richard Hammack's Elements of Discrete mathematics chapter 18 Prove that countable union of countable sets is countable.

Answers (1)

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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