Skip to main content

Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

Filter by
Sorted by
Tagged with
3 votes
1 answer
54 views

Proving that a set of three quadratic equations in three variables always has a solution

Given the following system of equations in $s, t, r$: $$ t^2 + s^2 - 2 t s \cos(\theta_1) = a^2 $$ $$ s^2 + r^2 - 2 r s \cos(\theta_2) = a^2 $$ $$ r^2 + t^2 - 2 r t \cos(\theta_3) = a^2 $$ How can I ...
Quadrics's user avatar
  • 24.4k
1 vote
2 answers
59 views

Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

I've recently been trying to understand some proofs about convex functions. The definition of convex I'm using is: Let $f(x)$ be a once differentiable function defined on $[a, b]$. $f$ is convex iff ...
Alp's user avatar
  • 409
-2 votes
2 answers
130 views

Is implication true if two statements are always the case?

I have a task that requires me to show that under a certain set of circumstances, a set has property A if and only if it has property B. I can show that under the given circumstances, the set always ...
ormondo's user avatar
  • 15
1 vote
1 answer
62 views

Choosing the Starting Point of Induction

I am a self-study student learning the basics of math proofs. During my studies, I encountered the following question: (For all n $\ge$ 3) I completed the problem using weak induction (with a ...
LateGameLank's user avatar
3 votes
0 answers
39 views

Permanent divisor in highly composite numbers

I was wondering whether it is true that if a particular divisor (be it prime or composite) appears for the first time in the sequence of highly composite numbers (HCNs), would it still be present for ...
BarbaraKwarc's user avatar
1 vote
1 answer
62 views

Prove that $c \sup A = \sup(cA)$ for $c>0$.

I'm new to real analysis and trying to prove $\sup⁡(cA)=c\sup⁡(A)$ for $c>0$. Using this definition of least upper bound: $s=\sup A$, where $s\in \mathbb R$ and $A\subseteq \mathbb R$ if $\forall ...
Ba_nanza's user avatar
  • 138
1 vote
0 answers
25 views

Proof about multiplying another submatrix while calculating the determinant [duplicate]

I took a linear algebra class. However, while lecturing on inverse matrix, the professor said that the following content is trivial, so We skipped over the proof. However, I was so curious about the ...
user1274233's user avatar
-1 votes
0 answers
29 views

product of structure constants [closed]

I'm not that familiar with Lie Algebra, however for a basis $F_{i} \in \mathfrak{su}(N)$ such that $$[F_{i}, F_{j} ] = \sum_{k}f_{ijk}F_{k}$$ I have seen it written that $f_{ikl}f_{jkl} = \delta_{ij}$....
LieAlgebraGuy1999's user avatar
0 votes
0 answers
15 views

Complete Lattices and the Injectivity of the Restriction $f|_S$ - Verification of Proof

Attempt (General Case) Conjecture: I want to show that if $X$ and $Y$ are nonempty sets, $(X, \leq)$ is a complete lattice, and $f: X \to Y$ is any well-defined function, then there exists a nonempty ...
Joshua Ortiz's user avatar
0 votes
0 answers
52 views

On the dimension of the zero set of a bounded linear functional on a Hilbert space.

Let $H$ be a complex Hilbert space and let $f\colon H \to \mathbb{C}$ be a linear bounded functional. Since $\ker f$ is a closed vector subspace of $H$ we have that $$H=\ker f\oplus \ (\ker f)^\perp.$$...
Jack J.'s user avatar
  • 1,076
0 votes
1 answer
95 views

If a contrapositive can be shown, then is a direct proof always possible?

If we can prove that $¬Y \implies ¬X,$ then is it always possible to prove that $X \implies Y$ without first proving that $¬Y \implies ¬X$ ? Motivation: when studying analysis, there are some problems/...
Red Banana's user avatar
  • 24.2k
0 votes
1 answer
34 views

Should I finish a proof assuming a (possibly not independent) additional axiom?

I am writing a proof and I do not know how to proceed. In the paper I’m writing, I have assumed four Axioms that I suspect imply one very particular thing. However, the proof contains a step that I ...
EoDmnFOr3q's user avatar
  • 1,222
9 votes
4 answers
956 views

Proof by Contradiction: "Bad Form" or "Finest Weapon"? Reconciling Perspectives [duplicate]

G.H. Hardy famously described proof by contradiction as "one of a mathematician's finest weapons." However, I've encountered claims that some schools of thought consider proof by ...
Nagaraju Chukkala's user avatar
1 vote
1 answer
54 views

Inductive Definition vs Inductive Proof vs Recursive Definition

I read some answers here that tried to explain the difference between "Inductive Definition" vs "Recursive Definition". But I couldn't really understand the difference between the ...
Junsu Kim's user avatar
  • 140
0 votes
0 answers
51 views

Is my proof strategy legit?

I am working on a proof that I am finding quite challenging. While I think I have been able to prove one of the many intermediate steps of said proof, the final approach to show that particular step ...
EoDmnFOr3q's user avatar
  • 1,222
1 vote
2 answers
113 views

Proving the fundemental theorem of Calculus

I am currently trying to write a proof for the fundemental theorem of Calculus, mainly to practice writing proofs, and I have an attempt below. The problem is, I am unsure of how to interpret the ...
Alice's user avatar
  • 508
0 votes
0 answers
27 views

Projection into a Hilbert space with respect to an orthonormal sequence.

Let $H$ be an Hilbert space and let $(e_k)_{k\in\mathbb{N}}$ be an orthonormal sequence in $H$. We define $$V:=\overline{\text{span}(e_k)}$$ I must prove that for all $x\in H$ the projection onto $V$ ...
FoxMath's user avatar
0 votes
0 answers
65 views

Proof by contradiction, where is the contradiction?

Example. Let $F=\{E_1,E_2,\ldots,E_s\}$ be a family of subsets with $r$ elements of some set $X$. Show that if the intersection of any $r+1$ (not necessarily distinct) sets in $F$ is nonempty, then ...
Shoe__gazer's user avatar
0 votes
0 answers
41 views

Double or triple induction proof

I have this equation: $$N_k(\left| V \right|,d) = \begin{cases} \begin{aligned} &1, & &\text{if } \left|V\right|= k \\ \end{aligned}\\ \begin{aligned} & 1+ \...
fib5555's user avatar
0 votes
1 answer
21 views

An alternative solution to the problem of sitting non-hostile ambassadors on a round table

I'm currently reading Problem-Solving strategies by Arthur Engel and came across the following question: $2n$ ambassadors are invited to a banquet. Every ambassador has at most $n−1$ enemies. Prove ...
Aryaan's user avatar
  • 281
2 votes
0 answers
85 views

Question about proof of Lebesgue Decomposition Theorem for the case of $\sigma$-finite positive measure.

I am asked to proof the following Lebesgue Decomposition Theorem for the case of $\sigma$-finite positive measure: Lebesgue Decomposition Theorem$\quad$ Let $(X,\mathscr{A})$ be a measurable space, ...
Beerus's user avatar
  • 2,483
0 votes
1 answer
44 views

For a square matrix $A$, if $\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$ * q ($\lambda$)

For a square matrix $A$, if $\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$ * q ($\lambda$) for some polynomial q. My approach: So we know that A has an ...
brodar's user avatar
  • 157
0 votes
0 answers
57 views

Characterization of the projection in a Hilbert space.

I have to prove that two variational problems are equivalent. Let $H$ be a complex Hilbert space and $K \subset H$ be a nonempty closed convex set. We know that $$y=P_K(x)\iff \begin{cases}\tag 1 y\...
Jack J.'s user avatar
  • 1,076
0 votes
0 answers
23 views

How to formalise this (simple) proof on chromatic polynomials

Theorem 2.2 Let G be a graph with n vertices. Then P(G, q) is a monic polynomial of degree n with zero constant term and with integer coefficients that alternate in sign. This is my current working, ...
Eshwar Kolli's user avatar
0 votes
1 answer
24 views

Proving an inequality with one constraint given

If $a,b,c>0$ and $a+b+c=1$ show that $6+3\left(\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b} \right) \le\frac{1}{a} + \frac{1}{b}+\frac{1}{c}$ My work : $6abc +3((ab)^2 + (bc)^2 +(ca)^2 ) \le ab+bc+...
the_bot_unknown's user avatar
1 vote
1 answer
35 views

Prove: If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.

I need to prove the following result: Let $(X,\mathscr{A},\mu)$ be a measure space. If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$. Here is my ...
Beerus's user avatar
  • 2,483
0 votes
3 answers
91 views

Proving inequality where $a^2 +b^2 +c^2 =3$ and $a,b,c$ are positive reals

For $a,b,c >0 $ and given $a^2 +b^2 +c^2 =3 $ prove the following inequality: $\large a+b+c +\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} \leqslant 3\left( \frac{a}{b} + \frac{b}{c} +\frac{c}{a} -1 \...
the_bot_unknown's user avatar
0 votes
0 answers
44 views

How does one prove this inclusion of a set?

I've recently come across the Cauchy's original proof for arithmetic-geometric inequality. In it, he does a proof by induction over a set $A$. Firstly he proves that $P_n\implies P_{2n}$ and then that ...
realreal's user avatar
  • 139
0 votes
2 answers
52 views

Prove the following inequality where the constraint is an inequality too

For $a,b,c>0$ and $a^2 +b^2 +c^2 \leqslant a+b+c$ prove the following inequality and show when we have equality : $F=\dfrac{1}{a^2 +b^2 +c^2 } +\frac{a^3}{b} +\frac{b^3 }{c} +\frac{c^3 }{a} \...
the_bot_unknown's user avatar
0 votes
1 answer
59 views

Maximum function recursive property

In an old book by Richard Bellman, I found a proof of the initial Bellman equation: $$ f_n(x)=\max_{0\le x_N\le x}\left[g_N(x_N)+f_{N-1}(x-x_N)\right] $$ where $x_i\ge 0$ and $\sum_{i=1}^Nx_i=x$. ...
TheKwiatek666's user avatar
0 votes
0 answers
22 views

Feedback on and assistance with this proof about a particular quotient space of $\mathbb{C}P^1$

The goal here is to define the particular equivalence relation I'm attempting to describe, and then provide an equation (in this case, (2)) that can be used to determine whether or not two given ...
Simon M's user avatar
  • 887
0 votes
1 answer
47 views

Prove that $L(X) \subseteq L(X \cup Y)$.

Given that $L(X)$ is a subspace generated by $X$, any vector in $L(X)$ is a linear combination of the vectors in $X$. That is: $$v=L(X)\rightarrow v=\sum_{i=1}^{m}a_i x_i: (x_i\in X, a_i \in R, \...
Antonius Anonymous's user avatar
0 votes
2 answers
83 views

Question on Theorem 1.11 in Baby Rudin

Definition 1.7 in Baby Rudin says: Suppose S is an ordered set, and E $\subset$ S. If there exists a $\beta \in$ S such that $x \leq \beta$ for every $x \in E$, we say E is bounded above and call $\...
Dr. J's user avatar
  • 149
1 vote
0 answers
100 views

On the absolute continuity of a particular functions.

We consider $f\in L^1[a,b]$, where $F$ is a finite interval of $\mathbb{R}$. The function $F\colon [a,b]\to\mathbb{R}$ defined as $$F(x):=\int_{[a,x]}f\;d\lambda\quad(x\in [a,b])$$ it is called ...
MathMister's user avatar
0 votes
0 answers
62 views

Equivalence of two cross product definitions

I've come across two different definitions of a cross product between vectors $\mathbf{a}$ and $\mathbf{b}$. Let $\mathbf{a} \land \mathbf{b}$ denote this vector cross product. The vector cross ...
DC2974's user avatar
  • 111
0 votes
1 answer
18 views

Proof of the "local property" of limits

I'm working with Calculus by Spivak and here's the Problem I'm currently solving: Suppose there is a $\delta > 0$ such that $f(x) = g(x)$ when $0 < |x - a| < \delta$. Prove that $\lim_{x\to ...
Aryaan's user avatar
  • 281
1 vote
2 answers
44 views

Conventions for expressing nested logical relations

Suppose I want to say that the conditional statement $P\implies Q$ is also equivalent to $(\sim P)\lor Q$. Writing it like this is confusing: $P\implies Q \equiv (\sim P)\lor Q$ I have a few solutions....
bluesky's user avatar
  • 225
7 votes
7 answers
274 views

Prove that the sequence defined by $a_{n + 1} = 2a_n^2 - 1$ is always negative iff $a_1 = -\frac{1}{2}$

I want to prove that the sequence defined by $a_{n + 1} = 2(a_n)^2 - 1$ is always negative if and only if $a_1 = -\dfrac{1}{2}$. Here are some observations I've made so far. However, I haven't been ...
Christopher Miller's user avatar
0 votes
4 answers
67 views

Is there a simple proof for this integral trick?

I recently learned that a integral of type $\int \frac {p x + q}{a x^{2}+ b x + c} \text {d} x$ can be written as $$px + q = A \cdot \frac{\text {d}}{\text {d} x} \left(a x^{2} + b x + c \right) + B$$ ...
altmathic's user avatar
29 votes
5 answers
5k views

Are there statements so self-evident that writing a proof for them is meaningless? Is this an example of one?

Context: I know nothing about proofs and only a small amount about formal logic used in proofs. I'm trying to learn the basics of how to write a proof. For example, suppose I wanted to prove that &...
matt_rule's user avatar
  • 419
0 votes
0 answers
144 views

Help me to verify the proof of this theorem, which is proving $k$ª $=$ $\prod_{i=1} ^{\infty}$ $\mathbf K_{p_i}$ by using maximality and minimality

I want to verify my proof is true or false. The exercise what I want to prove is under theorem. $\mathbf {Exercise}$: Let $\mathbf k$ be some perfect field and $\mathbf K_{p_i}$ is a compositum of ...
Snailman's user avatar
2 votes
1 answer
52 views

Question About Signed Measures

I am self-studying signed measure, and I come across the following construction: Let $\mu$ be a signed measure on the measurable space $(X,\mathscr{A})$, and let $A$ be a subset of $X$ that belongs ...
Beerus's user avatar
  • 2,483
2 votes
0 answers
57 views

How much notation should there be in a formal proof? Is there a general guideline?

I am writing a conference paper in formal language theory with an involved proof and I've found myself struggling with notation. In particular, I don't know when to favor notation and when to favor ...
JonasPK's user avatar
  • 31
0 votes
0 answers
23 views

How Does One Show That A R.V Has CDF F?

I'm self studying probability using Statistics 101 book. In chapter 3 there's a question(ex. 9): Let F1 and F2 be CDFs, 0 <p< 1, and F(x) = pF1(x) + (1 − p)F2(x) for all x. (a) Show directly ...
BigTittyHooka's user avatar
1 vote
0 answers
55 views

Let $I=(x^2+x+1)$. Is $\Bbb Z_3[x]/I$ an integral domain?

Let $I=(x^2+x+1)$. Is $\Bbb Z_3[x]/I$ an integral domain? My solution goes like this: If $\Bbb Z_3[x]/I$ is an integral domain then $I$ is a prime ideal. But $I$ is a non-zero prime ideal and since, ($...
Thomas Finley's user avatar
0 votes
0 answers
39 views

How do I prove this in symbolic logic if the premise is just a tautology?

I'm working on some proofs in symbolic logic and I pretty much get it, but I'm having an issue with the last one. Usually the premise sets up the proof and I can walk through it, but this one just ...
Daniel's user avatar
  • 71
0 votes
0 answers
21 views

Proving that function maximizer is the kth smallest value in list

Given positive integers $L_1\leq L_2 \leq \dots \leq L_n$, I'd like to prove that for any given $k=1,2,\dots,n$ the function $f(u,k)=ku - \sum_{j=1}^n\max\{u-L_j,0\}$ attains its maximum value for $u=...
Joris Kinable's user avatar
1 vote
1 answer
113 views

Prove that $ (1 + 2x)(1 - y)(1 - z) \geq (1 + 2(xyz)^{\frac{1}{3}})(1 - (xyz)^{\frac{1}{3}})^2 $ [closed]

I checked on WolframAlpha, and it seems that the inequality $(1 + 2x)(1 - y)(1 - z) \geq (1 + 2(xyz)^{\frac{1}{3}})(1 - (xyz)^{\frac{1}{3}})^2 $ holds true, given the constraints $0 < x \leq 0.5$, $...
ido kahana's user avatar
0 votes
0 answers
50 views

Using A Proof Assistant as a Learning Aid

I don't see that anyone has asked this before, but if I have missed something please let me know. tl;dr - Can you use a proof assistant (like Lean) as a tool to learn math rigorously, and improve ...
RudyJD's user avatar
  • 119
4 votes
5 answers
133 views

How can I prove that $|A - B| = |A| - |B|$ where $B\subseteq A$

I denote the cardinality of a set $S$ as $|S|$. Now, how do I show that if $B\subseteq A$, then $|A - B| = |A| - |B|$? For this proof, I tried induction on the cardinality of $B$. Suppose that $|B| = ...
Aryaan's user avatar
  • 281

1
2 3 4 5
318