# Questions tagged [formal-proofs]

For questions about proofs within a formal system, such as natural deduction or Hilbert system.

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### How do I formally prove a universal implication?

A textbook I am reading (Discrete Mathematics and its Applications by Rosen) went from introducing formal propositional and predicate logic (including popular rules of inference like Modus Ponens, ...
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### Using the Intermediate Value Theorem to prove the existence of a number$\;$

I'm having a bit of trouble with something most everyone might find trivial, and I feel rather silly asking, but here it goes. The premise is as follows: "Use the Intermediate Value Theorem to prove ...
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### What is the reason we usually don't use formal proofs in mathematics?

Is the reason for not using formal proofs very often in mathematics because it is usually too lengthy for a person to make such a proofs or the reason is that it is simply not possible for everything ...
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### Understanding ex falso quodlibet together with proof by contradiction in a Gentzen style ND Proof

I began studying some formal logic for possible future proof and type theory dives. I am at the very beginning, Gentzen style natural deductions. Some of these proof rules defies my intuition so I ...
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### How to use natural deduction to show $\neg (P \land Q) \vdash \neg P \lor \neg Q$?

How to use natural deduction to show $\lnot (P \land Q) \vdash \lnot P \lor \lnot Q$? I think I need to first assume $\neg(\neg P \lor \neg Q)$ and then find a contradiction but I cannot see how to do ...
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### Conditional Statements/ Implication statements within a proof specifically linear algebra

I am sort of new to the mathstackexchange so excuse me for any mistakes that I make while writing this post. I've been working on proofs and ran into a conflicted view of how to prove conditional ...
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### Proof with disjunctive conclusion

I'm after a natural deduction proof of the following sequent: (P & Q) → R : (P → R) ∨ (Q → R) The textbook I'm using says there is a 24 line proof, but the shortest I've managed is 29 lines. I'...
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### Prove that $(p \to q) \land (q \to r)$ is equivalent to $p \to r$

$(P\implies Q)\land(Q\implies R)$ is equivalent to $P\implies R$. Is this true? How to prove this directly, not using truth tables?
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### Proof of FOIL Modern Algebra

I am trying to work through Birkhoff's A Survey of Modern Algebra independently, but am having difficulty getting off the ground with the proofs based on laws, rules, etc. I come from mostly soft ...
I'm trying to provide a general proof for the following theorem. Let $0 < n < 1000$ be an integer. If the sum of the digits of $n$ is divisible by $9$, then $n$ is divisible by $9$. The book ...
Need help with the steps for natural deduction: P1. $(A \rightarrow B) \rightarrow (C \rightarrow A)$ P2. $A \wedge (C \leftrightarrow B)$ P3. $(A \lor C) \to (A \to B)$ \$\therefore ...