From a50fc0e90f7003becb04927e8ec44128312090c8 Mon Sep 17 00:00:00 2001 From: bluepython508 Date: Wed, 1 May 2024 17:42:29 +0100 Subject: [PATCH] . --- MA1511/sets.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/MA1511/sets.tex b/MA1511/sets.tex index 3d6e903..4085044 100644 --- a/MA1511/sets.tex +++ b/MA1511/sets.tex @@ -254,11 +254,11 @@ $+$ and $\times$ on $\Z/k$: \section*{Countable Sets} A set $X$ is finite if $\exists n \geq 0. $ a bijection $\{1, ...n\} \to X$ \\ Pigeonhole Principle: for finite $X$, any injective $f: X \to X$ is also surjective. \\ -$\N$ is infinite. Proof: $f: \N \to \N$ is trivially injective, and $\neg\exists x.~f(x) = 0$, and so not surjective. \\ +$\N$ is infinite. Proof: $f: \N \to \N $ defined by $ f(x) = x + 1$ is trivially injective, and $\neg\exists x.~f(x) = 0$, and so not surjective. \\ By the inverse of the Pigeonhole Principle, $\N$ is infinite.\\ A set $X$ is \emph{countably infinite} iff there exists a bijection $\N \to X$. \\ A set is \emph{countable} iff it is finite or countably infinite. \\ -Any subset of $\N$ is countable. Proof: Let $X \in \N$.\\ +Any subset of $\N$ is countable. Proof: Let $X \subseteq \N$.\\ If $X$ is finite, it's trivially countable. Otherwise, $X$ is infinite and it must be shown that $X$ is countably infinite. \\ For $k \in \N$, $X_{>k} = \{ n \in X | n > k \}$. Then $X_{>k} \not= \varnothing$, as $X$ would be a subset of $\{1..k\}$ and would be finite. \\