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0
votes
5answers
47 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
1answer
48 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 ...
2
votes
1answer
107 views

Need help to proof

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
34 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
44 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 ...
3
votes
1answer
70 views

$V=W_1\oplus\cdots\oplus W_k \iff \dim(V)=\sum{\dim(W_i)}$

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 ...
0
votes
0answers
39 views

property of an increasing or decreasing function

For $x \in \mathbb{R}$, and $f(x)$ an increasing function, can we prove whether $$ af(x)\lesseqgtr f(ax) $$ for $a >0$? If we have additional information that $f$ is homogeneous of some degree, ...
2
votes
4answers
147 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 ...
2
votes
0answers
90 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
88 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
46 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
39 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
85 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
49 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 ...
-6
votes
2answers
61 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
0answers
67 views

Best mathematical proofs [closed]

For you wich are the best mathematical proofs? I can remember Furstenberg´s proof of the infinitude of primes that really amaze me. I am very interested in this kind of proof that really can ...
2
votes
2answers
49 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
17 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
61 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
55 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
37 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
35 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 $\det{A} = 0$.
5
votes
1answer
39 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
38 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.
1
vote
0answers
37 views

Using Compactness to obtain an inequality.

I'm reading a proof that the Hausdorff dimension of the Cantor set is $\frac{\log 2}{\log 3}$ using the definition of Hausdorff dimension. The lower bound of the proof requires a lemma. (the problem ...
3
votes
1answer
63 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 ...
4
votes
2answers
147 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
46 views

Combinatorics identity sum of

Prove that: $$\sum^{n}_{k=0}\binom{k}{2n-k}2^k = 2^{2n}$$ By using only combinatorics identities.
0
votes
1answer
30 views

prove $f^{-1}(B)=A$

I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$ I want to prove that $f^{-1}(B)=A$. I just assume that here one is talking about ...
3
votes
2answers
120 views

Where is wrong in this proof [duplicate]

Suppose $a=b$. Multiplying by $a$ on both sides gives $a^2 = ab$. Then we subtract $b^2$ on both sides, and get $a^2-b^2 = ab-b^2$. Obviously, $(a-b)(a+b) = b(a-b)$, so dividing by $a - b$, we find ...
2
votes
1answer
22 views

Residue of a 1-form in a Riemann Surface does not depend of the chart

Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by $$ ...
0
votes
1answer
14 views

Clues to prove average in T is minor or equal than average in a smaller inner interval.

Suppose I want to prove (or disprove) this assertion Let $f$ be a discrete function, $T,h,k$ are constants So these terms are averages over $T$ and over $h$ $\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
7
votes
2answers
166 views

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$ $(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
0
votes
0answers
41 views

Proof contraction differentiable function

$g$ : $R$ $\rightarrow$ $R$ be a diferentiable function such that $-1$ < $a$ < $b$ < $0$ where for $y$ $\in$ $\Re$, $a$ $\le$ $g'(t)$ $\le$ $b $ Prove that $g(t) = t + f(t)$ is a contraction ...
1
vote
0answers
35 views

Proving that the circumcenter is the centroid

Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
1
vote
2answers
62 views

Help with proofs: Show that $AA^T$ and $A^TA$ are symmetric

I need help with a proof for my liner algebra class. If $A$ is a square matrix, then $AA^T$ and $A^TA$ are symmetric. I have no idea where to start!
3
votes
1answer
34 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
5
votes
3answers
112 views

Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]

The problem is following, prove that: $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$ I've tried solving this problem using mathematical induction, but I ...
2
votes
2answers
37 views

proof by induction to demonstrate all even Fibonacci numbers have indices divisible by 3

I am practicing proof by induction, and would like to use induction to prove the following hypothesis about the Fibonacci numbers: $$(\forall n\ge0) \space 0\equiv n\space mod \space 3 \iff 0 \equiv ...
1
vote
1answer
32 views

$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$

Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
2
votes
2answers
58 views

How do I prove the arithmetic-geometric mean inequality?

I am following along with this bare-bones proof of the arithmetic-geometric mean inequality with two real numbers. I'm having difficulty understanding the logic behind this step: $$ ...
3
votes
1answer
39 views

Rice's theorem_Theory of computation

Is there any body tell me, where is wrong in this proof Problem: The set of number of turing machine that has 5 state is decidable or not? Answer: The set is obviously 'Set of partial computable ...
4
votes
3answers
74 views

Extreme Value Theorem Proof (SPIVAK)

Them: If $f$ is continuous on $[a,b]$, then there is a $y$ in $[a,b]$ such that $f(y) \geq f(x)$ for each $x \in [a,b]$ OKay first of all how on earth does one come up with $g(x)$? It just ...
1
vote
1answer
51 views

Is this proof on the product of $X$ OK?

Let $X^2$ be star $\sigma$-compact and $F$ be a closed subset in $X^2$. If $\mathcal{U}$ is an open cover of $F$, then there exists a $\sigma$-compact subset $A$ of $X$, such that $F \subseteq ...
1
vote
1answer
41 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
2
votes
1answer
47 views

Are there examples of theorems proved via proper (i.e. non-conservative) extensions?

This is not a question about set theory specifically, but lets talk about ZFC just for concreteness Suppose we have a sentence $\phi$ in the language of ZFC, and a proof that either $(\mathrm{ZFC} ...
0
votes
1answer
38 views

Finding a reccurence relation for the following problem

A circular disk is cut into n distint sectors, each shaped liek a piece of pie and all meeting at the center point of the disk. Each sector is to be painted red, green, yellow, or blue in such a way ...
-2
votes
1answer
52 views

Given the following recurrence relation, prove using mathematical induction

How can we prove this using mathematical induction? $m_1 = 0$ $m_k = m_{\lfloor (k/2) \rfloor} + m_{\lceil (k/2) \rceil} + k-1$ for all integers $k \geq 1$ Prove using mathematical induction that ...
0
votes
6answers
111 views

Finding the number of subsets of S

How can we find the number of subsets of $S=\{1,2,3,...,10\}$ that contain neither 5 nor 6? Thanks!
0
votes
2answers
49 views

Use the binomial theorem to expand

How can we expand this using the binomial theorem? $(x^2 + 1/x)^7$

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