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Created: 2025-09-30
Type: Zettel
aliases:
References:
Links:
- "[[Countable & uncountable sets]]"
tags:
- MATH31AH
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> Definition: A set $A$ is called **infinite** if it has a proper subset $A_{1}$ which is bijective to $A$
> Definition: $A$ is a **proper subset** of $B$ ($A\subsetneq B$) if $B-A$ is nonempty (there is some element in $B$ that is not in $A$)
- If $A$ has $n$ elements, then any set $B$ bijective to $A$ also must have $n$ elements
> Example: Prove that $\mathbb{N}$ is an infinite set. Let $f: \mathbb{N} \to 3\mathbb{N}$. $3\mathbb{N} = \{ 3n:n \in \mathbb{N} \}$. This function is both injective and surjective, and is therefore bijective (each input maps perfectly to an output).
> $3\mathbb{N}$ is a proper subset of $\mathbb{N}$. For example, $1 \notin 3\mathbb{N}$
> Thus, $\mathbb{N}$ is infinite.
- Note that a [[Functions|function]] being [[Injective, surjective, bijective functions]] depend on the domain and the codomain. Changing them might change them might change their status.
