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

8
votes
1answer
248 views

Proving that $\omega(N)\neq4$ for an odd perfect number $N$ by hand

Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$, and let $N$ be an odd perfect number. It is not difficult to show that $\omega(N)\ge3$. In fact, Nocco already ...
2
votes
3answers
60 views

Prove Satisfiability of Property by Set

What is a proof strategy for proving that some property is satisfied by a particular set of numbers. For example, what would be an approach for proving that the archimedean property is satisfied by ...
1
vote
1answer
1k views

Suppose a,b are real numbers, if a is rational and ab is irrational, then b is irrational (Is my solution correct?)

Suppose $a,b$ are real numbers, if $a$ is rational and $ab$ is irrational, then $b$ is irrational. Solution: Proof by contraposition $$b = \frac{p}{q}$$ $$ a = \frac{j}{k}$$ where $p,q,j,k$ are ...
2
votes
3answers
631 views

Proving if $x$ is an even integer then $x^2 -6x +5$ is odd

If $x$ is an even integer, then $x^2 - 6x + 5$ is odd. My solution (direct proving): $$ x = 2k$$ $$ x^2 - 6x + 5 = 4k^2 -12k + 5 $$ $$ 4k^2 -12k + 4 + 1 = 2(2k^2-6k+2)+1$$ which is by definition is ...
1
vote
1answer
93 views

Is this pumping lemma solution correct?

Let Σ = {a, b, c}. Use the pumping lemma to prove that A = {aibicj | i,j ≥ 0} is not regular. Please make sure that your proof is clear, logical and complete. The solution that I wrote was: ...
2
votes
1answer
148 views

Approximate Nash Equilibrium

I am sort of confused by the notion of approximate Nash equilibrium. I will try to express my confusion in the following exercise. Problem. Is it true that for every two player game where every ...
-3
votes
1answer
93 views

$\operatorname{rank} (T^*) = \operatorname{rank} (T)$ : PROOF

Let $T$ be a linear operator on a finite dimensional inner product space. Prove that $\operatorname{rank}(T^*) = \operatorname{rank}(T).$
1
vote
0answers
62 views

Trying to prove an identity about a product

I have a product formula that, given a tuple $(a_1, \dots, a_{n-1}, 0)$, computes a dimension (of a vector space) via the following formula: $$ \mathrm{dimension} = \prod_{i < j} {(a_i + \cdots + ...
2
votes
1answer
65 views

Inner Product Spaces : $N(T^{\star}\circ T) = N(T)$ (A PROOF)

Let $T$ be a linear operator on an inner product space. I really just want a hint as to how prove that $N(T^{\dagger}\circ T) = N(T)$, where "$^\dagger$" stands for the conjugate transpose. Just as ...
4
votes
3answers
164 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
3
votes
3answers
348 views

Evaluate $\displaystyle\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
0
votes
1answer
97 views

Proving inequality of addition and multiplication

Which approach can be used to proof the inequality of the following equation? $$ax^2 + cy^2 = -2bxy.$$ We only know that $a > 0$ and $c > 0$ as well as: $$ac - b^2 > 0.$$ Therefore $b$ ...
6
votes
3answers
104 views

How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction

I've done the base step, but how do I prove it is true for $n+1$ without using a fallacy? $$4(n!)>2^{n+2}\quad \text{for } n\geq 4$$ Please help.
5
votes
1answer
60 views

Equivalence between these definitions of ordinal numbers

Von Neumann defines an ordinal $\alpha$ as a transitive set whose elements are well-ordered with respect to the membership relation $\in$. Meanwhile, in Naive set Theory, Halmos defines an ordinal ...
2
votes
3answers
74 views

How to prove/show $1- (\frac{2}{3})^{\epsilon} \geq \frac{\epsilon}{4}$, given $0 \leq \epsilon \leq 1$?

How to prove/show $1- (\frac{2}{3})^{\epsilon} \geq \frac{\epsilon}{4}$, given $0 \leq \epsilon \leq 1$? I found the inequality while reading a TCS paper, where this inequality was taken as a fact ...
1
vote
2answers
234 views

Proving a biconditional statement with an or

I want to prove a theorem in geometry of the form $p \iff q \vee r$. My plan is to prove: $q \implies p$ as well as $r \implies p$ $p \text{ and } \lnot q \implies r$ Can I get someone to verify ...
1
vote
1answer
132 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
3
votes
3answers
109 views

Multipliciousness within an inner product space.

Question: Let $V$ be an inner product space and $v,w\in V$. Prove that $\lvert\langle v,w\rangle\rvert=\lVert v\rVert \lVert w\rVert$ if and only if one of the vectors $v$ or $w$ is a multiple of ...
4
votes
2answers
4k views

This is a possible proof of the Riemann Hypothesis [closed]

http://arxiv.org/abs/1305.6845 The above link claims to have solved the Riemann Hypothesis. It's not mine, of course. I just saw this on Tumblr and realized I needed bigger guns. This proof looks like ...
1
vote
0answers
68 views

Proving $\text{Var}{(\hat{y}_h)} = \sigma^2 \left(\frac{1}{n} + \frac{(x_h-\bar{x})^2}{S_{xx}}\right)$

I have asked in another question how $\text{Var}{(\hat{y}_h)} = \sigma^2 \left(\frac{1}{n} + \frac{(x_h-\bar{x})^2}{S_{xx}}\right)$. Note that $\hat{y}_h$ = $b_0 + b_1X_h$ which is a regression line ...
5
votes
3answers
214 views

Help with Cartesian product subsets [duplicate]

I want to prove that if $A \subseteq C\,$ and $\,B \subseteq D,\,$ then $\,A \times B \subseteq C \times D.$ I know that $A \subseteq C \iff a \in A \rightarrow a \in C$ and that $B\subseteq D\iff ...
2
votes
2answers
398 views

How do I prove Binet's Formula? [duplicate]

My initial prompt is as follows: For $F_{0}=1$, $F_{1}=1$, and for $n\geq 1$, $F_{n+1}=F_{n}+F_{n-1}$. Prove for all $n\in \mathbb{N}$: ...
3
votes
2answers
244 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
6
votes
4answers
162 views

Is there a simpler approach to these system of equations?

I recently came across the following system of equations: $$x + y + z = 1 \\ x^2 + y^2 + z^2 = 2 \\ x^3 + y ^3 + z^3 = 3$$ And I have two questions: One, is there a way to prove or disprove ...
53
votes
15answers
7k views

Prove if $n^2$ is even, then $n$ is even.

I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that ...
0
votes
5answers
258 views

Proof of the equality of the difference of two sets iff sets are equal (direct vs. indirect)

I have a problem with the following (really) basic result: $$A\backslash B=B\backslash A \Longleftrightarrow A=B$$ More specifically, I am able to prove it only by contradiction (in particular in the ...
3
votes
2answers
323 views

Natural Deduction proof for $\forall x \neg A \implies \neg \exists xA$

$\forall x \neg A \implies \neg \exists xA$ I won't ask you to solve this for me, but can you please give some guiding lines on how to approach a proof in NDFOL? There are many tricks that the TA ...
3
votes
1answer
173 views

Need help to prove

I got the result below during my research. $$1=\frac{1}{1+a_1}+\frac{a_1}{(1+a_1)(1+a_2)}+\frac{a_1a_2}{(1+a_1)(1+a_2)(1+a_3)}+\frac{a_1a_2a_3}{(1+a_1)(1+a_2)(1+a_3)(1+a_4)}+... \tag 1$$ ...
2
votes
0answers
64 views

Proof for a finite number of elements

if I want to proof something for a restricted finite number of elements, meaning the following: Imagine that I have a theorem that is somehow similar to the following: For each element in ...
3
votes
2answers
57 views

Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,…k\}$?

I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a ...
4
votes
1answer
136 views

$V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{\dim(W_i)}$

If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
2
votes
4answers
291 views

Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?

I want to use Cantor's diagonalisation argument to prove that the set S of surjective functions of the form $\Bbb{N} \to \Bbb{N}$ is uncountable. The normal procedure is creating a matrix and filling ...
3
votes
0answers
207 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
3
votes
2answers
131 views

REVISTED$^1$: Circumstantial Proof: $P\implies Q \overset{?}{\implies} Q\implies P$

To prove that if a matrix $A\in M_{n\times n} ( F )$ has $n$ distinct eigenvalues, then $A$ is diagonalizable is enough to show that the opposite holds? That is, if $A$ is diagonalizable, then $A$ has ...
1
vote
1answer
79 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
3
votes
3answers
59 views

Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$

I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
4
votes
4answers
1k views

Proving one function is greater than another

How can I prove $f(x)$ $>$ $g(x)$ for all $x > 0$ given $f(x) = (x+1)^{2}$ and $g(x) = 4qx$ where $q$ is a constant in $(0, 1)$? My approach was to show that $(x+1)^2 > 4qx$ for the interval ...
0
votes
2answers
60 views

If $E = \{ x \in \mathbb{R}: \sin(\frac1{x}) = 1\}$ then $l = 0$ is a limit point of E

If $E = \{ x \in \mathbb{R}: \sin\left(\frac{1}{x}\right) = 1\}$, then $l = 0$ is a limit point of $E$. I have a proof here but I don't quite understand a few points, I hope someone can explain it a ...
-4
votes
2answers
99 views

Help with Theorem III.3.11 in Hungerford's algebra book

I need help to prove part (i) of this theorem which I couldn't prove. Any help would be appreciated. Thanks in advance.
2
votes
2answers
68 views

Suppose $f$ is a real-differentiable function on $[a,b]$ and suppose $f'(a)<c<f'(b)$. Prove then there is a point $x \in (a,b)$ such that $f'(x)=c$

This is what i have: Put $g(t) = f(t) - ct$. Then $g'(a)<0$ so that $g(t_{1}) < g(a)$ for some $t_{1} \in (a,b)$ and $g'(b)>0$ so that $g(t_{2}) < g(b)$ for some $t_{2} \in (a,b)$. ...
2
votes
1answer
51 views

Let $f$ be defined on $[a,b]$, Prove that if f has a local maximum at a point $x \in (a,b)$, and if $f'(x)$ exists, then $f'(x)=0$

Is this proof correct: Let's choose a $\delta$ to that $a < x - \delta < x < x + \delta < b$ If $ x - \delta < t < x$ then $\frac {f(t) - f(x)} {t-x} \geq 0$ Letting $t ...
4
votes
0answers
192 views

Good examples of proofs in mathematics exemplary of creative reasoning [closed]

Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
1
vote
1answer
61 views

Prove (without quoting any theorems) that polynomials on [0,1] are continous

I'm confused as to go about this problem. I feel as if we have to show that $P [0,1] \in C^{0}[0,1]$ by letting $f = a_{n}x^{n} + a_{n-1}x^{n-1} + .... + a_{1}x^{1} + a_{0}$ We must show that ...
0
votes
1answer
49 views

Bilinear Forms: An Initial Condition Proof

Let $B$ be a bilinear form on a finite dimensional vector space $V$. Suppose that for any nonzero vector $v \in V$ there exists a $w \in V$ such that $B(v, w)\neq 0$. Prove that for any linear ...
0
votes
2answers
70 views

Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices

Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
7
votes
1answer
2k views

$\inf A = -\sup(-A)$

Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$ So far this is what i have ...
1
vote
1answer
52 views

Prove that a polygon with nonnegative area is determined by at least three points.

How do you prove this statement in geometry? A polygon with nonnegative area can't be formed with fewer than 3 points.
3
votes
1answer
106 views

Proof the following trig series

Prove that $$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$ I am not necessarily looking for a ...
6
votes
2answers
414 views

Real Numbers is a subset of Complex Numbers?

So, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$ But, since the complex numbers' definition is $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$, doesn't that mean ...
1
vote
1answer
58 views

Combinatorics identity sum of

Prove that: $$\sum^{n}_{k=0}\binom{k}{2n-k}2^k = 2^{2n}$$ By using only combinatorics identities.