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Questions tagged [formal-proofs]

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

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

844 questions
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Is an axiom a proof?

From this comments discussion on Philosophy.SE: "Check out formal logic resources - I'm not going to dig them out for you. Alternatively ask on Math.SE. An 'axiom is a proof' is a definition in formal logic - and not an axiom. In philosophical…
Cort Ammon
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Proof that for any $16$ digit number there is at least one sequence of $1$ or more digits which its product is a perfect square

I came across this problem where one is asked to proof that, for any $16$ digit number there is at least a sequence of $1$ or more digits which its product is a perfect square. For example, in the number $$4,562,348,973,245,984$$ The product of…
Ioannes
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Proof that the square root of a non-perfect square is irrational

Is the reasoning for this proof logically sound? To show that the square roots of all non perfect squares are irrational, we must first know the definitions of rationality and perfect square. A number n is said to be rational if it can be expressed…
djackson5790
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Proof by contradiction - Getting my head around it

Hey there Math community! I have a general question on contradiction and it's getting difficult to get my head around it. Notes: I have some background in math and I have read several proofs by contradiction already For the sake of the argument,…
hargun3045
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Prove that, for any positive real numbers a, b, and c, $a^3b^2 + a^2b^3 + a^3c^2 + a^2c^3 + b^3c^2 + b^2c^3 \geq \frac{a^5 + b^5 + c^5}{5}$.

For this question I assume you would start with a rewritten form so that $a^3b^2 + a^2b^3 + a^3c^2 + a^2c^3 + b^3c^2 + b^2c^3$ = $(a^3 + b^3 + c^3)(a^2 + b^2 + c^2) - (a^5 + b^5 + c^5)$, but how would I progress from there to prove the inequality? I…
bob123
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Is there only one right solution when solving with Fitch system?

I was wondering if there was only one dead-set answer for each Fitch proofs. Are solutions that are given by the textbook the only way to answer a Fitch proof question (do all the steps have to look exactly the same?) ?
cluelesschloe
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What is missing on Gödel's theorem explanation in "Emperor's New Mind" from Roger Penrose?

In "Emperor's New Mind", from pages 132 to 141, Roger Penrose explains Gödel's incompleteness theorem in very simple terms. Basically he says that, in a system of ordered propositions ($P_1(w)$, $P_2(w)$, ..., $P_n(w)$) and ordered proofs ($\Pi_1$,…
fiatjaf
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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?
apostonaut
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Formal Proof Problem

I'm doing this formal proof problem with 10 steps, given these three premises; 1) (G V H) ⊃ I 2) (J V K) ⊃ ~I 3) K / ∴ ~H 4 - 10 is unknown. I tried using material implication and DeMorgan's Theorem on that step, but failed. Any help would be…
streetslicer
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Formal Proof of "No Largest Number"

I need to create a formal proof of there not being the largest number, with only the definition of "less than" being given with Peano arithmetic. $s(x)$ means successor of $x$ or $x + 1$. $$ \text{Premise}\\ \forall x\forall y (x \lt y \iff \exists…
Kevin
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Recommendation for first book to proof writing

I have a friend who is a third year physics undergrad. She wants to take her first proof based math course and she's decided on Abstract Algebra where the professor follows the book written by Dummit. What is a good book can she use for a month or…
Syed Bukhari
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for all sets A, B, and C if A ⊆ B then A$\cup$C ⊆ B$\cup$C.

So far I have this Suppose A, B, and C are any set where A $\subseteq$ B. Proof that A $\cup$ C $\subseteq$ B U C: (Show that x $\in$ B$\cup$C) Let x $\in$ A $\cup$ C and then by definition of union x $\in$ A or x $\in$ C. So by definition of…
Nolando
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prove that the $\sqrt{n}$ is unbounded

I wanted to check if my answer to proving that the $\sqrt{n}$ is unbounded works. If the $\sqrt{n}$ is bounded then there exists a $K$ s.t. $|\sqrt{n}|< K$, for all n. therefore $|\sqrt{n}| < K \Rightarrow -K < \sqrt{n} < K \Rightarrow (-K)^2 <…
ben
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Can we write formal (mechanical) proof of any theorem?

why formal proofs are not widely used? sometimes non formal proofs are cumbersome. are there any "important" theorems that have been proved formally
John Jack
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