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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1answer
24 views

Proof strategy for $(<=)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (e) $(<=)$ Assume that $g$ is one-to-one. Because $g$ is a ...
0
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2answers
28 views

Proof by induction sum $2^j = 2^{n+1} - 1$

I am trying to solve a previous test for an exam, and there are no solutions. The problem I am trying to solve is If $n$ is a natural number, then $1 + 2 + 2^2 + 2+3 + ... + 2^n = 2^{n+1} -1$ ...
4
votes
2answers
53 views

Linear Algebra: Direct Sum

Prove that if $W_1$ is any subspace of a finite-dimensional vector space $V$, then there exists a subspace $W_2$ of $V$ such that $V = W_1 \oplus W_2$ What I have done so far is to note that ...
2
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2answers
23 views

Help with real analysis proof involving supremum

Let $S\subseteq\Re$ be nonempty. Prove that if a number $u$ in $\Re$ has the properties: (i) for every $n\in N$ the number $u-1/n$ is not an upper bound of $S$, and (ii) for every number $n\in N$, the ...
-1
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2answers
37 views

Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
0
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2answers
22 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
1
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0answers
26 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
1
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2answers
76 views

How to prove a function from $\mathbb N\times \mathbb N$ to $\mathbb N$ is bijective. [duplicate]

I am having trouble with this problem: $f\colon \mathbb N\times \mathbb N \rightarrow \mathbb N$ is defined by $f(i,j)=\dfrac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection from ...
0
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1answer
18 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
0
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2answers
26 views

Proving that a set is denumerable without using a particular theorem

this question may seem like a duplicate of another one that I asked, but it is not. In this question, I am not allowed to use the Theorem which states: Every infinite subset of a denumerable set is ...
0
votes
1answer
17 views

Abstract Direct Product Proof Help

Let G = G1 x G2. Let H = {(x1, e2) : x1 ∈ G1} and K = {(e1, x2) : x2 ∈ G2}. (a) Prove H ≤ G and K ≤ G. (b) Prove that HK = KH = G (c) Prove that H ∩ K = {(e1, e2)} (d) Show that G/H is isomorphic to ...
0
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0answers
36 views

Proving that the set of irrational numbers is uncountable [duplicate]

Work: Assume that the set of irrational numbers is countable. Since $Q$ is infinite, it is therefore denumerable. Therefore, there exists a bijective function $f: N \rightarrow Q$. From here I am ...
0
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1answer
25 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
0
votes
1answer
22 views

Convex Subset Projection

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
0
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1answer
33 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
1
vote
1answer
35 views

Proving that $f: N\times N \rightarrow N$ is surjective [duplicate]

I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective. Work: I know that using the theorem in which $n$ is the product of prime numbers ...
0
votes
3answers
54 views

Induction, show that something is smaller then …

I have to show the following by induction. $1 \cdot 2 \cdot 3 ... (n - 1) \leq (\frac{n}{2})^{n -1}$ As it is homework I "only" need a push in the right direction. my thought is that is something ...
9
votes
4answers
344 views

Application of computers in higher mathematics

Currently the main application of computers in mathematics seems to be to compute things, i.e. to solve equations, evaluate integrals, etc. It is at all possible to delegate the thinking of a ...
0
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1answer
27 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
0
votes
4answers
51 views

Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
2
votes
2answers
34 views

Proving corollary to Euler's formula by induction

I'm currently looking at two proofs to the following corollary to Euler's formula and I'm not quite seeing how the authors can make a specific assumption in their proof. One proof comes from my ...
0
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0answers
7 views

Prove optimal schedule for $F \mid p_{ij} = p_j \mid \sum_j w_jC_j$

There's a homework exercise I have to do for my scheduling course which let me stuck. Consider $F \mid p_{ij} = p_j \mid \sum_j w_jC_j$. Assume that there are two jobs $j$ and $k$ such that ...
1
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1answer
29 views

$\epsilon - N$ proof confirmation.

These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it. My limit and required assumptions: ...
1
vote
1answer
47 views

What exactly does $\vdash_T G_T \leftrightarrow \lnot \exists y$ Prf$(\ulcorner G_T \urcorner, y)$ mean?

To me this translates to: $G_T$ is provable in $T$ if and only if there doesn't exist a $y$ such that $y$ is a witness to the provability of $\ulcorner G_T \urcorner$. But I'm not entirely sure what ...
1
vote
1answer
37 views

Prove that if graph $G$ is a 3-connected planar graph then its dual must be simple.

I'm trying to study for a quiz. I think I'm on the right track with this problem, however, I'm having a difficult time formalizing it. Prove that if graph $G$ is a 3-connected planar graph then its ...
0
votes
1answer
18 views

Prove the probability of n events intersecting

I'm trying to write a proof for this, but I don't know how to get started. Would proof by induction be the easiest way? If you could break it down into general steps I could wrap my head around, I ...
0
votes
0answers
8 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
0
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0answers
38 views

Number of roots of a polynomial (Proof)

What might be a simple proof to show that the maximum number of roots of a polynomial is equal to the degree of the polynomial? For example a quadratic polynomial can have a maximum of 2 roots. Can ...
0
votes
2answers
41 views

Minimize Function over Convex Subset

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
0
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0answers
14 views

Why $[\forall x \neg \alpha \rightarrow \neg \alpha^{x}_{c}] \longrightarrow [\alpha^{x}_{c} \rightarrow \exists x \alpha]$ is a tautology?

Let $c$ be a new constant symbol in the language. Then $[\forall x \neg \alpha \rightarrow \neg \alpha^{x}_{c}] \longrightarrow [\alpha^{x}_{c} \rightarrow \exists x \alpha]$ is a tautology. This ...
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1answer
28 views
0
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0answers
25 views

Lagrangian Method Proof

Suppose $f(\mathbf x)$, $g(\mathbf x)$ are smooth functions where $\mathbf x^*$ is a constrained local minimizer of $f(\mathbf x)$ subject to $g(\mathbf x)=0$. If $\nabla g(\mathbf x^*) \neq 0$ and ...
0
votes
0answers
9 views

Use specific choices of coefficients and find numerical evidence that the prices can oscillate wildly if the condition is not satisfied.

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
0
votes
2answers
16 views

If $c \not\equiv 0 \pmod p$ then $\forall a \not\equiv 0\ \exists b \not\equiv 1$ so $c+a\equiv ab \pmod p$

Im looking for a correct argumentation of why the folowing holds, any help would be great: For $p$ prime, if $c \not\equiv 0 \pmod p$ then $\forall a \not\equiv 0 \pmod p ~\exists b \not\equiv 1 ...
0
votes
0answers
23 views

A nowhere zero point in a linear mapping and Research Resources

Conjecture: If $\mathbb{F}$ is a finite field with at least 4 elements and $A$ is an invertible $n\times n$ matrix with entries in $\mathbb{F}$, then there are column vectors $x,y \in \mathbb{F^n}$ ...
0
votes
1answer
34 views

Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
1
vote
1answer
41 views

Convex Functions and Subsets

Suppose that $f, g: \mathbb R^n \to \mathbb R $ are $C^1$ convex functions. Show that $C = ${$\mathbf x \mid g(\mathbf x) \leq 0$} is a convex subset of $\mathbb R^n$. Show that if $\nabla f(\mathbf ...
0
votes
1answer
41 views

General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
1
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2answers
31 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
0
votes
2answers
39 views

Analysis: Prove the converse

It can be shown that if $\lim_{n\to\infty} a_n = L$, then $\lim_{n\to\infty} |a_n| = |L|$. Is the converse of this result true?
0
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0answers
25 views

Application of Compactness theorem

Let $\frak{R}$ be the structure $\left<\mathbb{N},+, \cdot,0,1\right>.$ If $\frak{A}$ is any structure for this language and if $a,b \in |\frak{A}|,$ we say $a$ divides $b$ in $|\frak{A}|$ if ...
0
votes
0answers
15 views

Order of a Group Help Abstract [duplicate]

Is the order of the Heisenberg group infinite since H = 1 a b 0 1 c 0 0 1 under matrix multiplication where a,b,c are real numbers? How would I formally state this?
0
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0answers
64 views

Prove that a certain graph and its dual are 4-colorable

Let $G$ be a simple planar graph with fewer than 12 faces. Suppose that each vertex of $G$ has degree at least $3$. prove that $G$ and its dual are 4-colorable. I'm not too sure how to approach ...
1
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1answer
35 views

We are interested in price of a commodity, traded at regular intervals. Why it is reasonable to take $a$, $c$, and $d > 0$ and $b < 0$?

We are interested in the price of a commodity which is traded at regular intervals. We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. ...
0
votes
0answers
24 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
1
vote
1answer
18 views

basic conditional probability proof

I having trouble with the following proof: $$P((A \cap B) \mid B) = P(A\mid B).$$ I get that $P(A\mid B) = P(A \cap B) / P (B)$, but I am unsure of how to proceed.
0
votes
4answers
35 views

Prove that for all integers n, $21 | (3n^7 + 7n^3 + 11n)$

I needed some help solving this. I know that we must show that it is divisible by 3 and 7 but how do I show that $$ 3n^7 + 7n^3 + 11n \equiv 0 \mod{3} $$
10
votes
1answer
92 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
-1
votes
0answers
40 views

Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [closed]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection $h:A\rightarrow ...
3
votes
1answer
31 views

Generalized Pythagorean triples construction.

All primitive Pythagorean triples $(a, b, c) : \{ a^2 + b^2 = c^2 \} \wedge \{ a \equiv 0 \pmod{2} \}$ can be expressed in the form:$$\{ a = 2pq, b = p^2 - q^2, c = p^2 + q^2 \}$$ for positive ...