## Undirected graph

### Set

context | $V,E$ … set |

definiendum | $ \langle V,\langle E,\psi\rangle\rangle \in \mathrm{it}(E,V) $ |

postulate | $ \psi $ … function |

postulate | $ \mathrm{dom}(\psi)=E $ |

postulate | $ \forall (e\in E).\ \exists (u,v\in V).\ \psi(e) = \{v,u\} $ |

### Discussion

In the above definition, the set $E=\{a,b,\dots\}$ in $\langle E,\psi\rangle$ is any set whos elements then each label an edge, e.g. $\psi(a)=\{v,w\}$.

Instead, one can also define a graph using a multiset $\langle E_\mathrm{ends},m\rangle$ where $E_\mathrm{ends}=\{\{v,w\},\{u,w\},\dots\}$ is itself a set of endpoints and $m:E_\mathrm{ends}\to\mathbb N$ counts the number of instances such a pair is part of the graph. The definitions are of course practically equivalent, the definition above with $\psi$ de-emphasises the focus on “$v$ and $w$ from $V$ are things which are connected” in favor of “$a$ is something from $E$ which connects the things $v$ and $w$ from $V$”.