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subsets of countable sets are countable
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(Corollary)
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The definition of countable sets would not serve us well if it did not conform with our intuition about countable sets. So let us prove that countable is in a sense hereditary.
Every subset of a countable set is itself countable.
Proof. Let $B\subseteq A$ and $A$ countable with $f:A\rightarrow K$ , $K\subseteq \mathbb{N}$ a bijective function as in the definition of countable sets.
Let us consider $f|_B$ , the function $f$ restricted to $B$ , i.e. $f|_B: B \rightarrow f(B)$ . Then $f|_B$ is trivially onto, but also one-to-one ( $f$ was one-to-one!). So we have a bijective function from $B$ onto $f(B)\subseteq K \subseteq \mathbb{N}$ , which completes the proof. 
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"subsets of countable sets are countable" is owned by beke.
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Cross-references: proof, one-to-one, onto, function, bijective function, subset, hereditary, countable
There are 2 references to this entry.
This is version 3 of subsets of countable sets are countable, born on 2006-03-14, modified 2007-04-15.
Object id is 7721, canonical name is SubsetsOfCountableSets.
Accessed 3152 times total.
Classification:
| AMS MSC: | 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers) |
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Pending Errata and Addenda
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