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Union of countable sets is countable
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[QUOTE="Bipolarity, post: 4387102, member: 176941"] [h2]Homework Statement [/h2] Prove that a finite union of countable sets is also countable. Is an infinite union of countable sets also countable?[h2]Homework Equations[/h2] A set S is countable if and only if there exists an injection from S to N.[h2]The Attempt at a Solution[/h2] I will attempt prove it for the case of 2 sets. Proving it for a finite collection of sets follows analogously. Suppose the countable sets are A and B. Then there are injections [itex] f_{A} [/itex] and [itex] f_{B} [/itex] from A to N and B to N respectively. We need to show the existence of an injection from {A+B} to N where + denotes union. Since {A+B} is the union of A and B, certainly it contains an element that is in at least A or in B (or in both A and B). Then each element of {A+B} has an injective mapping to N, since each element of {A+B} is in A or in B. Does this complete the proof? Is this rigorous? And what about the case for an infinite union? BiP [/QUOTE]
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Union of countable sets is countable
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