analogy: finite sets and finite dimensional vector space

Aspects

	finite set	finite dimensional vector space
bijection	Äquivalenz von Bijektivität, Surjektivität und Injektivität bei gleich mächtigen, endlichen Mengen	Äquivalenz von Isomorphismus, Monomorphismus und Epimorphismus für Vektorräume gleicher, endlicher Dimension
Restrictions	\(f: A \cong B\) bijection \(\Rightarrow \vert A \vert = \vert B \vert\)	\(\varphi: V \rightarrow W\) vs-iso \(\Rightarrow \mathrm{dim}(V) = \mathrm{dim}(W)\)
	\(\vert A \vert = \vert B \vert \Rightarrow\) existence of a bijection	\(\mathrm{dim}(V) = \mathrm{dim}(W) \Rightarrow\) existence of an vs-iso
	\(f: A \twoheadrightarrow B\) surjective \(\Rightarrow \vert A \vert \geq \vert B \vert\)	\(\varphi: V \rightarrow W\) vs-epi \(\Rightarrow \mathrm{dim}(V) \geq \mathrm{dim}(W)\)
	\(\vert A \vert \geq \vert B \vert, B \neq \emptyset \implies \exists f: A \twoheadrightarrow B\) surjective	\(\mathrm{dim}(V) \geq \mathrm{dim}(W) \implies \exists \varphi: V \rightarrow W\) vs-epi
	\(f: A \hookrightarrow B\) injective \(\Rightarrow \vert A \vert \leq \vert B \vert\)	\(\varphi: V \hookrightarrow W\) vs-mono \(\Rightarrow \mathrm{dim}(V) \leq \mathrm{dim}(W)\)
	\(\vert A \vert \leq \vert B \vert \implies \exists f: B \hookrightarrow A\) injective	\(\mathrm{dim}(V) \leq \mathrm{dim}(W) \implies \exists \varphi: V \rightarrow W\) vs-mono