Proof theory is an area of logic that studies proof as formal mathematical objects. For questions asking how to write proofs or for checking an informal proof, please use the proof-writing or proof-strategy tags instead.

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fake proof of $\forall a. \forall b. a = b \to 1 = 0$

I saw a less formal version of this fake proof that claimed to prove $2=1$ but because it assumed $a=b$ from the start I knew why it was wrong. It does seem however that the proof can be used to prove ...
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41 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
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54 views

prove that if n is odd then 5n +3 is even [closed]

I have been trying to find online tutorials but have been struggling. some help plus working out will be fully appreciated
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2answers
27 views

What does this proposition mean?

$∀x ∈ P(\Bbb{N}), x \notin \{\} \Rightarrow ∃y ∈ x, ∀z ∈ x \ | \ y < z$ Where $P(x)$ is the power set. I'm interpreting it as "in all subsets of the natural numbers, there exists a value smaller ...
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25 views

How to check homeomorphic embedding relation programmatically?

This is a follow up to this question and Deedlit's answer. I'm looking for a precise definition of the "hem?" (tree A homeomorphically embeddable in tree B?) relation, preferably in terms of a ...
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2answers
44 views

Help with understanding this proof in discrete mathematics?

This is the question and solution: Q: Prove that for any integer $a$, $2a + 1$ and $4a^2$ + 1 are relatively prime. A: Since $4a^2 + 1 = (2a − 1)(2a + 1) + 2$, any common divisor of $2a + 1$ and ...
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1answer
47 views

Proof by induction of this formula? [duplicate]

$2^0+2^1+2^2+...+2^n$ for $n ∈ \mathbb{N}$ U ${0}$. I made a conjecture that this is $2^{n+1} - 1$. Now I have to prove it by induction. I tested the base case where it's equal to zero, and it ...
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21 views

prove corollary to the Universal Non-Euclidean Theorem?

For any line l and any point P not on l, there are infinitely many lines through P parallel to l. In the proof of the theorem, the choice of point X on m uniquely determines the line PS. Show that ...
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1answer
23 views

Z / 6Z being a set of well dedfined equivalence classes, and a congruent to b(mod 6)

why is this = [0],[1],[2],[3],[4],[5],[6] and how would I define f Z/6Z - Z/6Z by f([a]) = ([2a]). I have the proof but I don't understand it. Proof: Assume [a1] = [a2] in Z/6Z. then a1 congruent to ...
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1answer
40 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
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1answer
20 views

Proving Theorem: subspace of polynomials of degree two or less?

How can I prove that the set $S$ of polynomials of degree $2$ or less, whose coefficients sum to zero, is a subspace of all polynomials with degree $2$ or less? I know I need to show that $a+b+c=0$ ...
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1answer
24 views

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$.

Define $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $f([a])=[3a+1]$. Prove that $f$ is well-defined, surjective and injective I don't really have a problem with figuring out if it's ...
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4answers
47 views

Proof by Induction Divisibility.

$6^n-5n+4$ is divisible by 5 for all positive integers $n$. $n >=1$ Prove By Induction My attempt is as follows: $n=1$ $6^1-5(1) +4$ $=5$, Therefore 5 is divisible by 5 so $n=1$ is true ...
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3answers
44 views

How can I prove that the square of an even number ends in 0/4/6?

I am trying to prove that the last digit of the square of an even number is either 0, 4, or 6 but I'm completely lost and have no idea how to tackle this problem.
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1answer
89 views

Uncountable reals in the theory

The Question I'm looking for a possibility to somehow proof the "essence" of Cantor's diagonal argument within a recursive first-order theory which is satisfied by the reals (better: within a theory ...
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0answers
28 views

How far can-I rewrite in lambda functions?

I am quite new with the lambda calculus. I am experimenting lambda-calculus proofs through the coq proof assistant, but the question I have is not related to coq (I guess). However, I'm going to use ...
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1answer
31 views

Euclid's proof for infinitely many prime numbers

Prove that there are infinitely many primes congruent to 3mod4 using euclid's proof for infinitely many prime number. I guess I don't really know where to start because I don't understand euclid's ...
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1answer
47 views

Comparing Statements and predicates using Truth Tables

Consider the four statements: $∃x$ $∀y$ $p(x, y)$ $∃y$ $∀x$ $p(x, y)$ $∀x$ $∃y$ $p(x, y)$ $∀y$ $∃x$ $p(x, y)$ which we call S1, S2, S3 and S4 respectively. Does there exist a predicate p such ...
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3answers
52 views

Proving $f\colon S \to S$; $f(x) = 1/x$ is bijective

Hey I'm trying to figure out this proof. I don't know if anyone could help but I would really appreciate it! Let $S = \mathbb{R} \setminus \{0\}$. Prove that the function $f\colon S \to S$; $f(x) = ...
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1answer
42 views

Model-theoretic question about language of field theory.

Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite ...
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1answer
53 views

Proof Strategy, Power sets, and Injections

Let $S$ and $T$ be sets and define the function $$f:\mathcal P(S) \times \mathcal P (T)\to \mathcal P(S \cup T)$$ by $f(A,B) = A \cup B$ for all $A \subseteq S$ and all $B \subseteq T$. Prove that ...
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How to find the shortest proof of a provable theorem?

Roughly speaking, there are some fundamental theorems in mathematics which have several proofs (e.g. Fundamental Theorem of Algebra), some short and some long. It is always an interesting question ...
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2answers
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how to know when a particular proof is appropriate for the given problem?

The main trouble I am currently having in math is knowing when the use cases are appropriate in a proof. I see many videos where they seem to choose a strategy like proof by contrapositive or proof by ...
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2answers
49 views

Undecidability and truth

Are there undecidable problems for which a single truth exists? For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds ...
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2answers
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Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom

I have no idea whether this question is way too specific or whether something similar has already been asked (we still need to work out a way to search for formulas I guess). Anyways here I go: I ...
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2answers
39 views

Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
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1answer
63 views

Is the parallelogram law a theorem or an axiom?

I'm learning about inner product spaces and I am able to prove it within an inner product space. Is this a theorem or an axiom in euclidean geometry?(note: not the geometry of Descartes)
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2answers
42 views

Trouble understanding algebra in induction proof

I'm on hour 20 of studying for the discrete math midterm tomorrow, and I've got to be honest I'm a little panicked. In particular I'm having trouble with induction proofs, not because I don't ...
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1answer
24 views

How can Goodstein's theorem be expressed in PA

I understand Goodstein's Theorem and its proof. I'm trying to understand the proof of why Goodstein's Theorem cannot be proved in PA. However, it's not immediately clear to me that Goodstein's Theorem ...
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2answers
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Proof by contradiction how to show is properly

For every $x \in \left[\pi/2,\pi\right]\,,\ \sin\left(x\right) − \cos\left(x\right) \geq 1$. I have drawn the graph and can clearly see that A is true however how do I prove it correctly.
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2answers
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If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

Let $n\in\mathbb{N}$. So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$. Help?
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1answer
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Euclidean algorithm to provde gcd's and multiples

Suppose a, b, n ∈ N. Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b). I was going to try setting it up, by literally doing: nb = rna + k and so forth, but something tells me this ...
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1answer
25 views

Proving the dimension of the intersection of 2 subspaces

Assume that $U$ and $W$ are distinct subspaces $( U ≠ W )$ of a four-dimensional vector space $V$ and $\dim(U) = \dim(W) = 3$. Prove that $\dim ( U ∩ W ) = 2$.
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How Do I Show that Condensed Derivable Rules of Inference Yield the Same Formula as Using Condendensed Detachment Multiple Times?

If we look at condensed detachment of two formulas $\alpha$ and $\beta$, we can see that D$\alpha$.$\beta$, where $\alpha$ has form C$\alpha$$_a$$\alpha$$_b$ is equivalent to using the rule ...
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1answer
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Why wouldn't someone accept Gentzen's consistency proof?

Reading the consistency section of the Peano Axioms wikipedia page, I came across this sentence: The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, ...
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Proof of law of reflection using Fermat's principle : are we really proving the law of reflection?

Before you skip reading this, let me tell you that this isn't a "how to derive the law of reflection using Fermat's principle" question. Also, I asked it on MSE instead of the physics site because ...
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0answers
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Prove that (√ 2)^(log(n)) + log^2(n) + n^10 = O(2^n) [duplicate]

This example question has me rather stumped. I'm not sure where to even start with the log(n) terms. The only clue it gives is "There is at least one non-trivial induction to do as part of the ...
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1answer
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Is there a proof for what I describe as the “recursive process of mathematical induction for testing divisibility”.

I was working on my homework for Discrete Math, and we were asked to "Prove: $6 | n^{3}+5n$,where $n\in \mathbb{N}$" my solution varied significantly from how I have seen it done by others. I noticed ...
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1answer
58 views

Question about the incompleteness proof (Theorem V)

Question in short: Where do I find a complete proof of Theorem V from Gödels incompleteness proof? If it does not exists, can someone provide it? Question in detail: I am trying to understand ...
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1answer
20 views

Proving Properties in Ordered Fields

Refer to Definition 1.3, which states, an ordered field is a field F that is ordered set with the following additional properties: ...
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2answers
75 views

Using logical Properties to prove a tautology

So I have to prove this as a tautology. I've been stuck on this forever and am not sure where to go. I experimented and got this far, and looking for some pointers on where to take it next. (p → q) ...
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1answer
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Birkhoff's completeness theorem

I have two simple questions. A) Does Birkhoff's completeness theorem follow directly from Gödel's completeness theorem? B) Is Birkhoff's completeness theorem constructive in the following sense: ...
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Has the abc conjuncture been proved by Shinichi Mochizuki? [duplicate]

I'd like to know is his proof was reviewed, and what exactly happened.
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30 views

Cartesian product proof with counterexample

I was asked to disprove the following statement by counterexample: Let A, B and C be sets. If A x C = B x C then A = B I was under the impression that: (x1, y1) = (x2, y2) if and only if x1 = x2 and ...
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1answer
18 views

Proving order of magnitude

Generally how much proof must be given to prove a statement of order-of-magnitude? for example: $n^2 + 2 log (n) = O(n^2)$ $2 log (n)$ has a lower order of magnitude than $n^2$ so it can be argued ...
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2answers
129 views

Is constructive proof of non-existence possible

Constructive proof construct(indicates) object that satisfies given predicate. Question is whether one can give constructive proof of non-existence of an object with given property e.g. that every ...
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1answer
47 views

Weakening and Contraction

I saw this site saying weakening is a structural rule where the hypotheses or conclusion of a sequent may be extended with additional members and that contraction is a rule where two equal (or ...
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57 views

What is really a “complete” deductive system for first-order theories.

Given some first-order language and a set of axioms therefrom one still needs to specify a deductive system to turn it into a full-fledged first-order theory. Currently I'm under the following ...
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Is a sentence in $\Pi_1$ true given $Q \vdash \lnot\varphi$?

If $Q \vdash \lnot\varphi$ (Q is the Robinson arithmetic), and if I assume that $\varphi \in \Pi_1$; Can I say that $\varphi$ is a true sentence? My thoughts are that, given that Q is $\Sigma_1$- ...
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1answer
69 views

A simple question on Gödel's functional interpretation

I've been recently reading the Gödel's functional interpretation (or Dialectica). It is generally defined inductively, as could be found here: http://www.andrew.cmu.edu/user/avigad/Papers/dialect.pdf ...