Answer by user1947180 for Can you explain to me why this proof by induction...
I think what is confusing you is that you are "starting out" with a graph of $n+1$ nodes.But note that you are not yet saying anything about its colorability. All you're trying to show here is that...
View ArticleAnswer by Dan Doel for Can you explain to me why this proof by induction is...
The proof you're citing doesn't have $P(n+1) → P(n)$ as the inductive step. It has $P(n) → P(n+1)$. It:Assumes $P(n) = $"all planar graphs with $n$ nodes are 6-colorable"Is given an arbitrary planar...
View ArticleAnswer by Hans Lundmark for Can you explain to me why this proof by induction...
They are not assuming $P(n+1)$ and using that to deduce $P(n)$, as you seem to believe.They are assuming that $P(n)$ is true (this is the induction hypothesis) and using this to deduce $P(n+1)$, by...
View ArticleCan you explain to me why this proof by induction is not flawed? (Domain is...
BackgroundI am following this MIT OCW course on mathematics for computer science.In one of the recitations they come to the below result:Official solutionTask:A planar graph is a graph that can be...
View ArticleAnswer by Pilcrow for Can you explain to me why this proof by induction is...
I think your confusion is ultimately about how you prove a universal sentence (each object of type X has property P) by induction.To prove a universal sentence, you consider an arbitrary object O of...
View ArticleAnswer by hunter for Can you explain to me why this proof by induction is not...
The specific flaw in your mock-proof (which I love, by the way) is that the property "there exists someone with at most 2 friends" is not stable under taking subsets. (Put differently, if we define a...
View ArticleAnswer by Douglas for Can you explain to me why this proof by induction is...
The missing step that I see in this proof is reasoning for why the subgraph, which is formed specifically by removing a degree <= 5 node, is guaranteed to still have a degree <= 5 node.Assuming...
View ArticleAnswer by ruakh for Can you explain to me why this proof by induction is not...
I think what's confusing you is that the theorem is indeed wrong, and the proof is missing a step!The theorem that you're asked to prove is "any graph can be colored in at most 6 colors." But that...
View ArticleAnswer by leftaroundabout for Can you explain to me why this proof by...
As rufflewind said, you intuition of inductive proofs as recursive programs is actually a good one and made precise by proof checkers like Coq, Lean or Agda. I won't go quite there, but would like to...
View ArticleAnswer by Rufflewind for Can you explain to me why this proof by induction is...
In the pseudocode you provided, a proof checker would complain about the second return statement: specialNode = findSomeDegree5orFewerNode(graph) subgraph = graph.drop(specialNode) return...
View ArticleAnswer by Ivan Aidun for Can you explain to me why this proof by induction is...
There are other good answers here, I wanted to offer a perspective that I think might appeal to a computer scientist: in a proof by induction, the inductive step is often presented as a way to build up...
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