subsets of countable sets are countable

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 countability is in a sense hereditary.

Theorem 1.

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 the proof.∎

单词	SubsetsOfCountableSetsAreCountable
释义	subsets of countable sets are countable 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 countability is in a sense hereditary. Theorem 1. 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 the proof.∎
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