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

Chardonnay Felix 2021-02-22 Answered
Discrete mathematics cardinality using Richard Hammacks
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tabuordg
Answered 2021-02-23 Author has 99 answers

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=SnnN for all n in N denote T, denote the set of all injections from SntoN. since Sn is countable so Tn is non empty. so, there exist a sequence tnTn define, f:SN×N be the mapping defined by f(x)=(n,tn(x)) where n is the smallest unique natural number such that x in Sn 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×N is countable so, there exist a: N×NN since composite of injections is injection thus, a circ f:SN is an injection therefore, S is countable

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