Fix notation

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. \\