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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(Can this be done in polynomial time?)Proving two graphs are isomorphic - Bondy/Murty - Graph Theory Page 6

I am trying to do the below problem: Now I can't see how one does this. I know you can explicitly show the bijections, but I can't see an easy way to do this, since they all have degree $3$(and ...
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1answer
18 views

Application of Riemann mapping theorem

Let $\Omega \neq \mathbb{C}, \emptyset$ be a simply connected domain and $a \in \Omega.$ Let $f:\Omega \to \mathbb{D}$ be a conformal map such that $f(a)=0, f'(a)>0.$ Could anyone advise me how to ...
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2answers
22 views

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ in some neighborhood of $x_0$

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ for all $x$ some neighborhood of $x_0$. My attempt is below. From the assumptions above, we have that $f(x_0) > M = f(x_1)$ for ...
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2answers
20 views

Even function divided by Odd function is an Odd function PROOF?

An Even function divided by Odd function is an Odd function,that is a fact. However is there a means to prove this?
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3answers
28 views

Is this formula satisfiable?

I am confused whether or not my explanation for whether or not this formula is satisfiable is correct. Note that the question state it should be Brief and it should not be necessary to write down a ...
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3answers
49 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
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0answers
7 views

Finding a Mobius transformation

Let $R=\{z\in \mathbb{C}: Re(z)>0, |z-3|>1\}$ and $A=\{z \in \mathbb{C}: 1<|z|<p\}.$ Find a Mobius transformation $f$ and $p$ such that $f$ maps $R$ conformally to $A.$ May I verify if my ...
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0answers
9 views

Showing that a collection of m solutions is linearly independant

Show that a collection $ \Phi_1 .. \Phi_m $ : I-->R of continuous functions satisfying $ \\ $ $ \int_I(\Phi_J(t)\Phi_k(t)dt $ =1 when j=k , 0 when j$\neq$k $ \\ $ is linearly independent. Multiply the ...
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1answer
13 views

Help needed to establish a conformal mapping

Could anyone advise me on how to find a conformal map from $H=\{z \in \mathbb{C}: Re(z)>0\}$ to $A= \{z \in \mathbb{C}:|z|>1, |z-2|<3\} \ ?$ I tried to compose the map in terms of ...
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2answers
36 views

Number of particles at time $t$

A following problem appears in my text book under the section of induction: At time $0$, a particle resides at the point $0$ on the real line. Within $1$ second, it divides into $2$ particles that ...
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3answers
43 views

List one of the ways in which Mario could buy the stars and comets. Note: Mario needs to spend all of his gold coins

Mario has 773500 gold coins to purchase a number of stars and comets. Each star costs 299 gold coins, and each comet costs 208 gold coins. If the number of stars that Mario buys is at least twice the ...
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2answers
46 views

Prove: $f(x)=e^{ax}$ is continuous on $\mathbb{R}$

Am I being fooled by how simple this statement looks? My book is currently telling me to take both $\lim_{x\rightarrow 0} f(x) =1$ and $f(x_1+x_2)=f(x_1)f(x_2)$ where $-\infty<x_1,x_2<\infty$, ...
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0answers
25 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
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2answers
33 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
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1answer
15 views

Proof verification for $f$ & $g$ surjective implies $fg$ surjective - Cohn - Classic Algebra Page 15

Question: Is this a valid proof? Side question: Am I less likely to get answers based on using notation $xfg=g(f(x))$? I want to prove that if $f$ and $g$ are surjective, then $fg$ is ...
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2answers
31 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
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1answer
34 views

Question on Proofs of Sets. [on hold]

The set $A$ is a subset of the set $B$ iff $A \cup B = B$ If $A$ is a subset of the set $B$, then $A \cup C$ is a subset of $B \cup C$.
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2answers
66 views

About the rationality of $1.1010010001\dots$ [duplicate]

Let's define $\rho=1.1010010001\dots$ which can be expressed by: ...
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4answers
77 views

For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a ...
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1answer
22 views

Proof verification: (Pretty pictures :) )Showing two graphs are not isomorphic. Bondy/Murty - Graph theory Page 5

I want to show the below two graphs are not isomorphic. Treat the left as graph $\bf G$ and the right as graph $\bf H$. $\bf G$ and $\bf H$ are not isomorhpic: Although we have a bijective mapping ...
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2answers
20 views

If $G$ is simple, then $\epsilon \leq {v \choose 2}$ - Bondy/Murty - Graph Theory with Applications Page 4

Question: Does this proof hold? Is this a bad proof? Any nicer proofs that don't rely on other theorems? Notation: $\epsilon$ - Number of edges $v$ - Number of vertices G - Here, any Graph ...
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1answer
17 views

The number of ways to paint a red tile in a grid.

here's the question: "You have nine tiles arranged into a three by three square mosaic. If you color each tile red or blue with equal probability, what is the probability that there exists a two by ...
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2answers
28 views

Induction on the number of marbles in a heap.

Here is the problem in full: "A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
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3answers
224 views

Show that the function is not continuous anywhere

I'm trying to prove that a specific function $f$ is not continuous for any $x_0$ that it is defined for. Here's what I have so far. Let $$f(x) = \left\{ \begin{array}{lr} -1 & : x\ ...
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1answer
37 views

Proving integration formulas from scratch

Prove the following integration formulas from scratch? (I uploaded them)
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3answers
52 views

Prove that $\sqrt[4]{1+y^4} \leq 1+|y|$

Prove that $\sqrt[4]{1+y^4} \leq 1+|y|$ for all real values of $y$. I attempted to show this by finding the power series expansion of $\sqrt[4]{1+y^4} $ and then relating that to $1+|y|$; however, I ...
0
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1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
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0answers
17 views

A question about a change of variable

I have came across this question while trying to find the derivate of the inverse functioin. And I have found the following limit: $$ \lim_{y\to y_0} = \frac{1}{\frac{f(x) - f(x_0)}{x-x0}}$$ We also ...
3
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3answers
270 views

Show that inequality holds

How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too? $a^2 + b^2 + c^2 \ge ab + bc + ca$ for all positive integers a, b, c I ...
0
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2answers
18 views

Proof by contradiction how to show is properly

For every $x \in \left[\pi/2,\pi\right]\,,\ \sin\left(x\right) − \cos\left(x\right) \geq 1$. I have drawn the graph and can clearly see that A is true however how do I prove it correctly.
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1answer
36 views

Calculus Proof involving exponents.

Prove that $2015^{2013}<2014^{2014}<2013^{2015}$ without the use of a calculator. I don't know where to begin here. Any help or guidance on where to begin would be greatly appreciated.
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5answers
69 views

For every natural number $n$, $\gcd(an,bn)=n\gcd(a,b).$

For every natural number I am trying to show that $n$, $\gcd(an,bn)=n\gcd(a,b).$ Here is my attempt. Put $d = gcd(a,b)$; we can write $d=aT+bJ$, where $T$ adn $J$ are integers. Then as $d|a$ and ...
0
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1answer
22 views

Vector spaces and direct sums

The map that was constructed in lectures is: $V,W$ subspaces of $U$. $f\colon V \oplus W \to U$ by the formula: $f((v,w))=v+w$ for $v$ in $V$, $w$ in $W$ Is it correct to generalise this to, ...
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1answer
41 views

Proof that for any $16$ digit number there is at least one sequence of $1$ or more digits which its product is a perfect square

I came across this problem where one is asked to proof that, for any $16$ digit number there is at least a sequence of $1$ or more digits which its product is a perfect square. For example, in the ...
0
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1answer
23 views

Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b).

Assume that a,b,n are all natural numbers. I was going to set it up as: na = q(1)*n(b) + r(1) where a>b and go down the chain: nb = q2 * r(1) + (r2) but something seems off. Someone told me ...
0
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1answer
24 views

Prove that if a|c and b|c, and a and b are relatively prime, than ab|c

How do I show this? I have an idea of what to do, but the problem overall is a little confusing to me. I can start the problem, but I just do not see how to get to the solution.
2
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1answer
30 views

Proof by induction regarding maximum number of questions one can ask.

sorry for the pretty ambiguous title. It's otherwise hard to describe this problem without stating it in full. There are $n$ points drawn on a whiteboard. Between every pair of points $X$ and $Y$ ...
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3answers
32 views

The second derivative of $f^{-1}$ and another question. :)

Suppose both $f$ and $f^{-1}$ are twice differentiable functions. Derive a formula for $(f^{-1})''$. My attempt: We have that by the inverse function theorem that: ...
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1answer
41 views

Use Fundamental Theorem of Arithmetic to prove that if $a >1$, $p$ is prime, and $p|a ^n$ for some $n \in \mathbb{N}$, then $p|a$

So, by the FTOA, since $a >1$, then a can be broken down into a product of a prime factors, so $a = p_1 \times p_2 \times \dotsm \times p_k$. Then, can I say that since $a$ is multiplied by itself ...
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2answers
41 views

Assume that 495 divides the integer 273x49y5 where x,y ∈ {0,1,2…9}. Find x and y.

So, I know that $495 = 5\times 9\times 11$. So then, if that's the case, then the number $\overline{273x49y5}$ must be divisible by $495$ if and only if it is divisible by 5 and 9 and 11. Then, I ...
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0answers
25 views

Prove the continuity and differentiability of a function in a point. [duplicate]

This question Is the same the question as this one (that I have posted yesterday at 12 am that I why I disconected from a large period of time) Prove that a function is both differentiable and ...
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2answers
43 views

Let n ∈ ℕ. If the sum of the digits of n is equal to the sum of the digits of 5n, then prove that 9|n.

I know how to test the divisibility of a number by 9, but only if I am given what n is. How would I set this problem up?
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1answer
49 views

Prove $2n+3 \le 2^n$ for all integers $n \ge 4$.

I have already started the problem but I am unsure on how to proceed. Prove $2n+3 \le 2^n$ for all integers $n \ge 4$. Base Case: Choose $n = 4$. $2n + 3 \le 2^n$ $2(4) + 3 \le 2^4$ $8 + 3 \le ...
0
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1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
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6answers
217 views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
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0answers
10 views

Proving relations are orders

The Problem Let P and Q be posets with respect to some order $\sqsubseteq$. Proof that the following relations are indeed orders. If P' is a subset of P, then P' is also a poset with ...
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2answers
122 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
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3answers
69 views

If $x^2$ is divisible by $4$ then $x$ is even?

I am studying discrete mathematics as course and I have to prove this "If $x^2$ is divisible by $4$ then $x$ is even". I am wondering how to prove it using the contrapositive of this ...
5
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3answers
126 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
1
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1answer
29 views

GCD Proof Questions?

Question 1: $a$ divides $b$ iff $d=a$, where $d=\gcd(a,b)$ My Solution: $d=at+bj$. If $a$ divides $b$ then $b=aK$. So $d=at+aKj$ hence $d=aN$. Plus as $d$ is the $\gcd$ $d=a$. And then in this method ...