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 documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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