For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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3
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1answer
47 views

Prove Lagrange's Identity without induction

Prove Lagrange's Identity without induction. $$ \sum_{1\leq j <k\leq n}(a_jb_k-a_kb_j)^2=\left( \sum_{k=1}^na_k^2 \right)\left( \sum_{k=1}^n b_k^2 \right)-\left( \sum_{k=1}^na_kb_k \right)^2 $$ I ...
0
votes
1answer
20 views

The Change-making problem algorithm proof (at the dynamic programming method)

I saw here the algorithm for the "Change-making problem" (at the dynamic programming method). I saw it here: http://www.columbia.edu/~cs2035/courses/csor4231.F07/dynamic.pdf I'm trying to find a ...
0
votes
0answers
26 views

Help with this solution of an exercise of Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution of the question 4.17 on page 97 (see the link above). What I didn't understand is why $V(J_z)$ are exactly those ...
0
votes
0answers
15 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
2
votes
3answers
43 views

Given a specific trapezoid, prove it is a rectangle

In quadrilateral ABCD, AB is parallel to CD. AC and BD meet at E. Points M and N are the midpoints of AE and DE, respectively. BM and BE trisect angle ABC, and CE and CN trisect angle BCD. Prove ...
1
vote
2answers
33 views

proof of laplacian $1/\rho$ in cylindrical coordinates at $\rho = 0$

In spherical coordinates, it can be proved that the laplacian of $r = \sqrt{x^2+y^2+z^2}$ at the origin is \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi\delta(\vec{r}) \end{align} as demonstrated in ...
2
votes
0answers
26 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
20
votes
10answers
415 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. For example: When I read from the book Proof from the Book, I saw there were ...
4
votes
1answer
30 views

Proof in Fulton's *Algebraic Curves*

I'm reading Fulton's algebraic curves book on page 106 and I didn't understand this proof: I didn't understand why can we assume $F_Y\neq 0$? (what $F$ irreducible has to do with this?). ...
0
votes
0answers
30 views

Proofs needed for certain results related to functional equations

Today our maths teacher told us the following results without stating the proofs: (These are all polynomial or exponential functions) 1) $f(x+y)=f(x)+f(y)$ then $f(x)=kx$ 2) $f(x+y)=f(x)f(y)$ then ...
1
vote
2answers
28 views

Leibniz Formula, proof of alternating property

$$F_{A} := \sum_{\sigma\in S_n}\operatorname{sign}(\sigma) \prod_{i=1}^n A_{i \sigma(i)}$$ I am trying the prove that $\det(A)=F(A)$. I know that to do this, I need to show that $F$ satisfies the ...
0
votes
1answer
26 views

Proving that $f(x)=0\ \forall x\in B(0,r)$

Let $y=f(x_1,…,x_n)$ be differentiable on $B(0,r)$. Assume that $\dfrac{\partial}{\partial x_i}f(x)=0\ \forall x\in B(0,r)$ and $i\in\{1,…,n\}$. How to prove that $f(x)=0\ \forall x\in B(0,r)$? Do ...
1
vote
1answer
20 views

Proving that the ball is converx

I need to prove that the ball $B(x,r)=\{y\in \mathbb{R^n}:||y-x||<r\}$ is convex. How to do this?
1
vote
1answer
32 views

Help to understand this proof in Fulton's book

I'm reading Fulton's algebraic curves book on page 105 and I didn't understand this proof: 1.Why if $R=k[X_1,\ldots,X_n]$, then $\Omega_k(R)$ is generated (as R-módule) by the differentials ...
1
vote
2answers
50 views

Find the lowest degree of the polynom $P$?

I have to determine the lowest degree of $P$ given by the following system : $\left\{ \begin{array}{l} P \equiv 2X \ \mod[X^2 -2X +1] \\ P \equiv 3X \ \mod[X^2 -4X+4] \end{array} \right.$ First, ...
0
votes
1answer
60 views

Proving set identities

I am attempting to work on some proofs for my math assignment, but I'll be honest in that I am really struggling to understand them. I read through the power point given by my teacher; however, even ...
2
votes
2answers
118 views

(Revisited$_2$) Injectivity Relies on The Existence of an Onto Function Mapping Back to Its Preimage

QUEST: For any sets $X$ and $Y$, there exists an injective function $f:X\rightarrow Y$ if and only if there exists a surjective function $g:Y\rightarrow X$. QUESTION$_1$: How do you people ...
0
votes
1answer
13 views

Heptagon (septagon) diagonal intersection

Diagonals TV and UW of regular heptagon TUVWXYZ meet at A. Prove that TU+TA=TW (Source: AoPS ItG). My observations: TUVW is an isosceles trapezoid Triangle TUA is congruent to triangle WVA
2
votes
1answer
24 views

Partial mapping, how does the inverse look like?

Let $X_1,X_2,Y$ be topological spaces. Let $f:X_1\times X_2 \to Y$ be continous at $a=(a_1,a_2)$. Show that the partial mappings $f_1:X_1\to Y; x\mapsto f_1(x) = f(x,a_2)$ is continous at $a_1$ and ...
1
vote
2answers
16 views

Will decreasing the variance of a subset, the global variance also decrease and vice versa? [on hold]

While implementing one of our propose algorithm we are assuming that, by decreasing the variance of a subset, the global variance will also decrease and vice versa. Is it something we need to proof ...
0
votes
2answers
2k views

Proof — Infinitely many primes of the form $4k + 3$ — origin of $4(p_1…p_k - 1) + 3$

I know there are sundry questions — like this pdf — and this (10.) Prove that any positive integer of the form $4k + 3$ must have a prime factor of the same form. Because $4k + 3 = 2(2k + 1) + 1$, ...
0
votes
2answers
38 views

Finite fields, characteristics and the Fundamental Homomorphism Theorem

I am trying to make sense of this proposition. I am fine with part (a), for part (b) however, can you explain what the computation proves? Can you not verify a homomorphism by checking the 3 ...
0
votes
1answer
27 views

Why $N= max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [on hold]

Using the proof development strategy used regarding the proposition (for all $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n - L| < \epsilon$ for all $n > N $) ...
4
votes
1answer
29 views

Prove: If $(\exists V $ neighbourhood of $ x)(f_{\mid V}:V\to Y$ is continous) then $f$ is continous in $x$.

Let $f: (X,\tau_X) \to (Y,\tau_Y), x\in X$ and $v\subseteq X$; Prove: If $(\exists V $ neighbourhood of $ x)(f_{\mid V}:V\to Y$ is continous in $x$) then $f$ is continous in $x$. I get ...
0
votes
1answer
35 views

Help me with this solution of the exercise 4.17 from Fulton's Algebraic Curves

I'm studying Fulton's algebraic curves book and in order to prove the well-definiteness of the divisor $div(z)$ on page 97 I'm trying to understand this solution which I found online. I didn't ...
0
votes
0answers
51 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
0
votes
0answers
17 views

Approximation of large unitary matrices

I have a hunch, that for now is only motivated by a physical argument. Could anyone point me in the right direction for proving (or disproving) it? The hunch is as follows: a unitary matrix $U$ of ...
1
vote
1answer
17 views

Can I assume a condition in the consequent?

Im reading Axler's Linear Algebra Done Right. In an exercise, he ask to prove that $$a\in F,v\in V,av=0 \implies a=0 \lor v=0 $$ where $V$ is a vector space over the field $F$. I've proved it this ...
-1
votes
0answers
8 views

Consistency and inconsistency of linear equations .

Can you verify the conditions required for the consistency and inconsistency for a pair of linear equations.
3
votes
2answers
31 views

proving a function as surjective

How can I prove a function is surjective? In the function $f: \Bbb{R}\to \Bbb{R}$, $$f(x) = 4x+7$$ we take $x = y-\frac{7}{4}$ and show that $f(x)=y$. How can this method prove that this function is ...
-2
votes
3answers
33 views

Prove/ Disprove; if a is divisible by bc, then a is not divisible by b and a is not divisible by c [on hold]

The way I am currently trying is using the contrapositive, so $a\mid b$ or $a\mid c$ $\implies$ $a\nmid bc$ so I am not sure how to prove this
0
votes
1answer
26 views

Understanding density of irrational numbers and Archemedian property

From Density of irrationals I know this much of the proof of the density of irrational numbers "We know that $y-x>0$. By the Archimedean property, there exists a positive integer $n$ such ...
1
vote
1answer
26 views

How to prove this inference in sequent calculus?

I'm using the event-B prover to proove some proof obligations. I have a relation representing a $table: table \in 1‥n \to \mathbb{N}$. I know that in a sorted table the following property is true: ...
0
votes
1answer
47 views

Show that there exists a unique function with a certain property

I'm trying to prove the following theorem: "Let $~f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}~$ be a function, and let $~c~$ be a natural number. Show that there exists a unique function $~a: ...
37
votes
9answers
7k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
2
votes
2answers
48 views

Proving $\log n < \sqrt n$

I am trying to prove $\exists n_0 > 0: \forall n > n_0: \log n < \sqrt n$. My attempt uses the series representation of the exponential function, but it does not seem to accomplish the proof: ...
4
votes
2answers
68 views

Help with a proof of pairs of real numbers

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a > b$ and $c > d$ with $a \neq c$ or $b \neq d$, such that $$\left|\frac{a-b}{c-d} - 1\right| < \frac ...
2
votes
1answer
22 views

Side-angle-side and side-angle-angle as proved by Euclid in the Elements (Proposition 26)

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html To prove that one side is equal to another, Euclid assumes that one side is bigger ...
0
votes
2answers
24 views

Understanding Proof that $\mathbb{R} \setminus A$ is dense. Verify proof.

Here's the proof I was given but with two minor? differences Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense. Proof: Suppose otherwise, then there exists real numbers $a$ ...
0
votes
2answers
48 views

If A is countable $\mathbb{R} \setminus A$ is dense. Clarify one line in proof? Ways to improve?

Here's the proof I was given: Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense. Proof: Suppose otherwise, then there exists real numbers $a$ and $b$, with $a < b$, such ...
1
vote
0answers
20 views

Developing proof writing and logical skills

What resources can a person turn to in order to develop their proof writing and logical skills? The advanced calculus course I'm taking has made me realize how weak my logic and proof writing skills ...
7
votes
3answers
7k views

Basic Linear Algebra Proof - Orthogonal Vectors

Prove that if $\mathbf{u}$ and $\mathbf{v}$ are nonzero orthogonal vectors in $\Bbb R^n$ they are linearly Independent. I've struggled with this a bit, here is what I know so far: Suppose ...
0
votes
0answers
24 views

Explain why $I$ is a function from $P$ to $P$ and determine whether it is one-to-one and onto.

The question and the solution are:( uploaded a photo so it is easier to see the formulas) So I am confused about the formula of p(x). P is the set of polynomial of x. OK, but why it makes p(x) = ...
1
vote
1answer
24 views

Lagrangians independent of $x$

In PDE Evans, 2nd edition, the following formula is printed as equation $\text{(9)}$ in §8.6 (on page 514): $$\sum_{k=1}^n (L_{p_i}u_{x_k}-L\delta_{ik})_{x_i}=0 \quad (k=1,\ldots,n) \tag{9}$$ ...
0
votes
1answer
19 views

Integrality conditions and proof by double counting.

Theorem $\mathbf{3.4.}$ In a block design of type $2-(v,k,\lambda)$ every element lies in precisely $r$ blocks, where $$r(k-1)=\lambda(v-1)\textit{ and }bk=vr\;.$$ The letter $r$ stands for ...
2
votes
0answers
31 views

How should I learn the Mathematical Proofs?

S.E advisers, What is the most efficient way to learn the basic proof methodologies, which are essential for studying the mathematical analysis and number theory? I am very interested in studying ...
-1
votes
1answer
29 views

Prove that: If $n$ is not divisible by $5$, then $n^2$ is not divisible by $5$ [on hold]

Suppose that $n$ is not divisible by $5$, then $n^2$ is not divisible by $5$. I tried using contrapositive to prove this, but I don't know how to proceed.
1
vote
2answers
35 views

Understanding line of given proof

I have to understand a set of proofs and I don't understand the reasoning behind this line "This is an injection, if $g(b_1) = g(b_2)$ then $F_{b_1}$ And $F_{b_2}$ intersect, which we shown never ...
2
votes
1answer
25 views

Proving a collection of subsets is a basis

I am given this definition of a basis: Let $a$ be a point in a metric space $X$. A collection, $\mathfrak{B}_a$, of neighborhoods of $a$ is called a basis for the neighborhood system at $a$ if every ...
1
vote
3answers
43 views

Prove that $\Gamma\left(-a\right)=\left[\Gamma\left(a\right)\right]^{-1}$ for $\Gamma:\mathbb{Z}\rightarrow \mathcal{B}\left(A,A\right)$

I am working through various problems in Bloch's Proofs and Fundamentals and I'm stuck on this problem (in need of hints): Let $A$ be a set. A $\mathbb{Z}$-action on $A$ is a function ...