MHB Prim's algorithm for minimal spanning tree

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Prim's algorithm is a method for finding the minimum spanning tree of a connected, undirected graph. To implement the algorithm, start with a single vertex and repeatedly add the smallest edge connecting a vertex in the tree to a vertex outside the tree until all vertices are included. Documentation of each step can include noting the vertices added, the edges selected, and the current state of the tree after each addition. The algorithm can still be applied to any connected graph, ensuring that it maintains the properties of a spanning tree. Understanding the definition of a connected spanning subgraph is crucial for correctly applying Prim's algorithm.
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Quite stuck on this how do i do this and how do i document each step?
 

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I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...

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