13. Countability

Sets being Finite / Infinite

Let $A$ be a set

1. $A$ is finite if $A=\varnothing$ or there exists $n\in N$ with bijection $f:A\to \{1,\cdots ,n\}$
2. $A$ is infinite if it is not finite

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Properties of Sets

1. Subset of a finite set is finite
2. Non-empty finite subset of $\mathbb{R}$ is bounded above
3. $\mathbb{N}$ is not bounded above (by the Archimedean property)

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Types of Countability of Sets

Let $A$ be a set then $A$ is

1. Countably infinite if there is a bijection $f:A\to \mathbb{N}$
2. Countable if there is an injection $f:A\to \mathbb{N}$
3. Uncountable if $A$ is not countable

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Countable Property of Sets

Let $A$ be a set

1. $A$ is countable if and only if $A$ is countably infinite or finite
2. If there is an injection $f:A\to B$ and an injection $g:B\to A$ then
   there exists bijection $h:A\to B$

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