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.

learn more… | top users | synonyms

0
votes
5answers
431 views

Prove: if x is odd, then sqrt(x) is odd.

If $x$ is odd, then $\sqrt{x}$ is odd, where $x$ is an integer. Any hints welcome and preferred. Thank you!
0
votes
1answer
41 views

Logistic Regression derivation

From the Wikipedia article http://en.wikipedia.org/wiki/Multinomial_logistic_regression: $ln \frac{\Pr(Y_i=1)}{\Pr(Y_i=K)} = \beta_1 \cdot \mathbf{X}_i $ $ln \frac{\Pr(Y_i=2)}{\Pr(Y_i=K)} = ...
0
votes
1answer
60 views

show analytic function such $f(z)={\operatorname{Log}(z+5)\over z^2+3z+2}$

Show that $f(z)=\dfrac{\operatorname{Log}(z+5)}{z^2+3z+2}$ is analytic everywhere except at the point $-1,-2$ and on the ray $\{(x,y):x\le -5,y=0\}$. i think that separate denominator and ...
3
votes
2answers
85 views

Passing a derivative through a limit.

After searching around on the net and on SE I have not found a satisfactory answer. Let $f_n: D \to \mathbb R$ be a sequence of functions. What assumptions, aside from $f$ being differentiable, do we ...
2
votes
0answers
48 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
0
votes
2answers
73 views

Suppose $F$ and $G$ are families of sets.

Suppose $F$ and $G$ are families of sets. Prove that $\bigcup F$ and $\bigcup G$ and are disjoint iff for all $A∈F$ and $B∈G$ , $A$ and $B$ are disjoint. It has been suggested to use contrapositive ...
1
vote
3answers
35 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
2
votes
1answer
65 views

Help Needed Showing that $\chi(\overline{G \times H}) \leq \chi(\overline{G}) \times \chi(\overline{H})$

Where $\chi(G)$ denotes the chromatic number, $\overline{G}$ the graph complement, and $\times$ the Cartesian Graph Product: I need to show that $(\forall G,H)( \chi(\overline{G \times H}) \leq ...
3
votes
3answers
74 views

Is it possible to use mathematical induction to prove a statement concerning all real numbers, not necessarily just the integers? [duplicate]

I am referring to the part of proof by mathematical induction where you show that "if it is true for one value k then it is true for the value k+1". Does proof by induction work over all real numbers? ...
1
vote
2answers
36 views

Proofs involving Well-Defined and One-to-One

Chartrand, 3rd Ed, P224-225: Define a relation $R$ as a relation from A to B. $R$ is well-defined means: $(a,b), (a,c) \in R \implies b = c$. P220: A function $f: A \to B$ is one-to-one ...
1
vote
0answers
40 views

How to Devise Combinatorial Arguments for Proving Identities

What are some strategies or tips for contriving/devising combinatorial arguments? Combinatorial proof for $\binom{n}{a}\binom{a}{k}\binom{n-a}{b-k} = \binom{n}{b}\binom{b}{k}\binom{n-b}{a-k}$? I ...
0
votes
1answer
50 views

Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...
0
votes
1answer
29 views

Prove that the distance function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ $0<p<1$ is a metric on R^n

Hi I am trying to prove that for $0<p<1$ the function $d_p(x,y)=\sum_1^n |x_i-y_i|^p$ is a metric on $\mathbb{R}^n$. I am struggling with the triangle inequality part; We have to prove ...
0
votes
2answers
38 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
2
votes
2answers
88 views

Proving a math statement is true or false

For all $x$ there is a $y$ such that if $x$ is non negative then $y^2 = x$ Is my logic correct in proving that statement is true ? Can provide an explanation of how to test this proof ? $x=2$ ...
5
votes
6answers
887 views

Positive integers expressable as sums of powers of 2

I need to prove that any positive integer is expressable as $$x=2^{j_0}+2^{j_1}+2^{j_2}+...+2^{j_m}$$ where $m\ge 0$ and $0\le j_0\lt j_1\lt j_2\lt ... \lt j_m. $ I think I get the gist of the proof; ...
1
vote
3answers
49 views

Disproving 0 as a dividend

Prove each of the following statements. (a) For all $b \in \mathbb{Z}$ if for all $k \in \mathbb{N}$, $b \not\mid k$, then $b = 0$. By hypothesis: $b \not\mid k \implies b\ell \neq k, \ell ...
3
votes
1answer
62 views

How to show that $P_n(x)$ have n distinct roots [duplicate]

Let $P_n : \mathbb R \to \mathbb R $ , $n\in \mathbb N$ be defined by $P_n(x)=\frac{\displaystyle1}{\displaystyle2^n n!}\frac{\displaystyle d^n}{\displaystyle dx^n}[(x^2-1)^n]$ I need to show that ...
0
votes
1answer
78 views

Converse of DIC

I am looking to prove the converse of the Divisibility of Integer Combination. I know how to prove the contrapositive of this statement but not the converse ... any help? Below is my attempt at ...
2
votes
6answers
108 views

Investigating the linearity between squares and their roots

I recently noticed that $\sqrt{128} = 11.31$ and that $128$ is $\approx 30\%$ between $121 = 11^2$ and $144=12^2$, that is: $$ \frac{128-121}{144-121} = \frac{7}{23} \approx 30\%$$ and $\sqrt{128} = ...
2
votes
1answer
129 views

How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
1
vote
1answer
156 views

Find bases given that P is the change of coordinates matrix from this to this [Lay P244 Q4.7.19]

Lay P289: Let $V$ be an $n$-dimensional vector space, let $W$ be an $m$-dimensional vector space, and let $T$ be any linear transformation from $V$ to $W$. To associate a matrix with $T$, choose ...
0
votes
1answer
14 views

proof of uniqueness (hint) $f(m,n)=n+1$ only if $n \ge m$

I'm a bit confused about this proof. let define on $\Bbb N$ a binary fucntion $f$ that satisfies (1) $f(m,n)=n+1$ if $n \ge m$ and (2) $f$ is commutative if I write the values of $f$ in a 2x2 ...
0
votes
3answers
51 views

Strategy to solve inequalities

I want to prove $|x + y| = |x| + |y|$ iff $xy \ge 0$. I don't understand a place to start. I was thinking of solving using contradiction but, as I am new to real analysis, I don't understand it. ...
0
votes
1answer
69 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
2
votes
1answer
102 views

Curl Proof Question

Prove the given formula. So far I have $f\textbf{F}=(f\textbf{F}_1, f\textbf{F}_2, f\textbf{F}_3)$, but I'm not sure where to go from there. Could anyone give me some pointers? Thank you.
1
vote
0answers
37 views

Gradient Proof Question

Prove the given formula ($r=||{\textbf{r}}||$ is the length of the position vector field $\textbf{r}(x,y,z)=x\textbf{i}+y\textbf{j}+z\textbf{k}$). $$\nabla \dfrac{1}{r} = \dfrac{-\textbf{r}}{r^3}$$ ...
8
votes
2answers
279 views

An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$

Let $c\in\mathbb{R}\setminus\{ 1\}$, $c>0$. Let $U_i = \left\lbrace U_{i, 0}, U_{i, 1}, \dots \right\rbrace$, $U_i\in\mathbb{R}^\mathbb{N}$. We know that ...
0
votes
1answer
79 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
0
votes
0answers
33 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
2
votes
1answer
56 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
0
votes
2answers
33 views

Proof relating to the order of $a \mod n$?

The proof required is to show that $\operatorname{ord}_n(a^j) \mid\operatorname{ord}_n(a)$, for any positive integer $j$. I have considered using a proof by contradiction, but am having trouble going ...
0
votes
0answers
29 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
0
votes
0answers
68 views

series-parallel system reliability equation proof

I am trying to device a proof for the series-parallel system reliability equation. The mathematical form of the equation is as follows: $Pr(\bigcap_{i=1}^{N}\bigcup_{j=1}^{M} A_{ij}(t)) = ...
1
vote
1answer
17 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
1answer
27 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
0
votes
2answers
31 views

Different way showing a subgroup is a subgroup of another subgroup

http://crazyproject.wordpress.com/2010/04/11/subgroups-and-quotient-groups-of-solvable-groups-are-solvable/ Lemma 1: Let $G$ be a group and let $H,K,N \leq G$ with $N$ normal in $H$. Then $N \cap K$ ...
0
votes
1answer
22 views

How do I work out the last sentence in this section of a proof of the Unique Factorization Theorem?

The last sentence states that the number of possibilities is $2\log_2 n$ (see the below image to follow the proof). I don't understand how to get $2\log_2 n$ but I understand everything that comes ...
3
votes
2answers
116 views

Proving Injectivity $x + \sin(x)$

I'm trying to prove injectivity of a particular function (without calculus), but I've come across a bit of a problem. The function is: $$f(x) = x+\sin(x)$$ I started by (abiding by common standards) ...
0
votes
2answers
36 views

How to prove something at Normal distribution

$X\sim N(\mu,\sigma^2)$. $A,B$ are constants and $A\ne0$. How to prove that $AX+B\sim N(A\mu+B,A^2\sigma^2)$ ? Thank you!
0
votes
5answers
89 views

Is proof by algorithm credible?

I have found this question How to prove that the inverse of a matrix is unique? And while the accepted answer is fine I was wondering if it's possible to proof the uniqueness by algorithm. There is ...
1
vote
1answer
43 views

Prove $A(A+B)^{-1}B = B(A+B)^{-1}A$ where $(A+B)^{-1}$ and $A$ and $B$ are $n\times n$ matricies.

Hi I have been working on this problem for the longest time. Prove: $ A(A+B)^{-1}B = B(A + B)^{-1}A$ We know that A & B exist in real space, and that they are also N x N matrices. It is also ...
2
votes
1answer
73 views

Do we sometimes have to go “each way” separately for iff proofs?

So, I often enjoy trying to prove "if and only if" statements by only using if and only if arguments. i.e. RTP: $A \Leftrightarrow D$. Proof: $A \Leftrightarrow B \Leftrightarrow C \Leftrightarrow ...
0
votes
3answers
30 views

Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
1
vote
2answers
71 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
2
votes
2answers
48 views

How to Complete Sketch of a function of two variables $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ ? [Stewart P930 Question 14.7.4]

For $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ $\implies$ $\partial_x f = 3 - 3x^2, \partial_y f = -4y + 4y^3$. Set both equations to 0 $\implies x = \pm $1 and $y = 0, \pm 1$. $1.$ To determine the ...
1
vote
1answer
87 views

Sketch Saddle Point of a function of two variables $ f(x, y) = 4 + x^3 + y^3 - 3xy$ [Stewart P930 Question 14.7.3]

As regards $ f(x, y) = 4 + x^3 + y^3 - 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I'm not asking about this, and am asking only about sketching contours. $1.$ ...
2
votes
1answer
52 views

Prove that these Sets Containing Infinitely Many Incompressible Strings Exist

We define a set $A$ to be special if: $$\liminf_{n \to \infty} \frac{|A^{\leq n}|}{n} = 0$$ I want to prove that there are special recursive sets that contain infinitely many incompressible strings. ...
1
vote
1answer
181 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
0
votes
1answer
28 views

T/F prove for modified Ramsey's theorem

By Ramsey's theorem we know that: $\forall k \in \mathbb N : \exists N \in \mathbb N$ that an arbitrary graph $G$ on a set of vertices $\{1,2,...,N\}$ contains $k$ vertices, which represent either a ...