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Mathematics
General Math
Math Proof Training and Practice
Direct Proofs regarding Induction
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[QUOTE="CGandC, post: 6390136, member: 620168"] [B]Summary::[/B] . [U]When asked to prove by Induction, I'm asked to prove a statement of the form:[/U] Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating the above statement to the following statement: ## P(0) \land \forall n ( P(n) \rightarrow P(n+1) ) \rightarrow \forall n ( P(n) ) ## And this is what they are proving. And eventually they say " therefore, it was proven that ## \forall n ( P(n) ) ## by induction " However, I don't really understand this. This is why: Let's denote the statement ## \forall n ( P(n) ) ## as ## W ##. To prove ## \forall n ( P(n) ) ## by induction means to prove ## W ## . Let's denote the statement ## P(0) \land \forall n ( P(n) \rightarrow P(n+1) ) ## as ## Q ##. To prove ## P(0) \land \forall n ( P(n) \rightarrow P(n+1) ) \rightarrow \forall n ( P(n) ) ## means to prove ## Q \rightarrow W ## Now's the problem: If we prove ## Q \rightarrow W ## then we only prove this implication, we are not proving ## W ##. Meaning, we are only proving ## P(0) \land \forall n ( P(n) \rightarrow P(n+1) ) \rightarrow \forall n ( P(n) ) ## . And not ## \forall n ( P(n) ) ## So how can it be that the problem of proving ## \forall n ( P(n) ) ## is translated into the problem ## P(0) \land \forall n ( P(n) \rightarrow P(n+1) ) \rightarrow \forall n P(n) ## ? Because they are both different statements and thus proving these are different proofs. [/QUOTE]
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Mathematics
General Math
Math Proof Training and Practice
Direct Proofs regarding Induction
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