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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2
votes
2answers
81 views

If $x$ is an integer then $x^2+ 5x - 1$ is odd.

What would be a proof strategy for this? I would like to show a proof of the contrapositive: if the expression is not odd, then $x$ is not an integer. If I go that route, how do I express the ...
2
votes
2answers
75 views

formula for the $n$th derivative of $e^{-1/x^2}$

$f(x) = \begin{cases} e^{-1/x^2} & \text{ if } x \ne 0 \\ 0 & \text{ if } x = 0 \end{cases}$ so $\displaystyle f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} ...
3
votes
0answers
30 views

Proof of equilateral triangle given angles

Let's say we start with a scalene triangle ABC, with no given angle measures or side lengths: Then, we add 3 Isosceles triangles adjacent to this one, given that they have angle measures ...
0
votes
2answers
35 views

Proof or a counterexample of a function

I have the following exercise, how can I proceed? Let $A$ and $B$ be sets, with $S \subset A$ and $f:A\to B$ a function, and $g:A\to B$ be an extension of $f\rvert_S$ to $A$. Does $g$ equal $f$? ...
2
votes
1answer
52 views

What is satisfiable by a reduct of a model is satisfiable by the original model (and vice versa)?

My professor told me that any formula that is satisfiable by a reduct of a model is satisfiable by the model it is a reduct of, and vice versa (as long as the formula is interpretable on the ...
4
votes
2answers
117 views

What is a good approach to demonstrate solvability of this type of puzzle without use of brute-force?

I chanced upon this puzzle in this question on the Anime & Manga site, and, like the OP, tried to solve it without any success. Here is a representation of the puzzle: the blocks may only be moved ...
0
votes
0answers
29 views

Help with solving ODE differently [duplicate]

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
1
vote
1answer
32 views

Contradiction in proof that in an integral domain, every prime is irreducible.

Let $\pi$ be a prime element in an integral domain. So, $\pi$ is a non-unit and if $\pi \mid ab \ $ then $\pi \mid a$ or $\pi \mid b$. An irreducible element $z$ is an element such that if $z=ab$, ...
0
votes
0answers
20 views

Locally injective function is globally injective [duplicate]

Let $f:\mathbb R\to \mathbb R$ be a continuous: Is the next statement true? If $f$ is locally injective for every real $x$ then $f$ is globally injective in $\mathbb R$ I think this theorem is true: ...
0
votes
2answers
33 views

Difference between open sets and open balls in metric space

Let $X$ be a separable metric space and let $\mathfrak{M}$ be the $\sigma$-algebra generated by open balls in $X$. Show that $\mathfrak{M}$ contains all the open sets in $X$ and all the closed ...
3
votes
3answers
42 views

Proving that $8^n-2^n$ is a multiple of $6$ for all $n\geq 0$ by induction

I have the following induction problem: $8^n-2^n$ is a multiple of $6$ for all integers $n\geq 0$. So far this is what I've done: Base case: $n = 0$ $8^0-2^0 = 6$ $1 - 1 = 6$ $0 = 6$ This ...
0
votes
2answers
65 views

Approaching this proof problem? If $0 \le x \le 3$ then $12 - 7x + x^2 \ge 0.$

Prove that if $x$ is a real number in the range $12 - 7x + x^2 \ge 0.$ Which type of proof should I use to solve this? At first I thought direct proof. Choosing a number between $0$ and $3$ and ...
4
votes
1answer
217 views

Prove that there exists an $n\in\mathbb{Z}\cup\left\{-\infty,+\infty\right\}$ such that… (Dynamics)

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the following dynamics described by $T\colon X\to X$ as follows: A 1 becomes a 2, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its two ...
0
votes
2answers
13 views

finding the derivative of w'Bw with respect to w

How should I approach to proving the following? $d(w^TBw)/dw = 2Bw$ where $B$ is a symmetrical matrix and $w$ is a vector.
2
votes
0answers
21 views

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length.

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length. I believe this is not hard problem, but I just do not have an idea how to start. I tried to ...
3
votes
4answers
139 views

Prove that $371\cdots 1$ is not prime.

Prove that $371\cdots 1$ is not prime. I tried mathematical induction in order to prove this, but I am stuck. My partial answer: To be proved is that $37\underbrace{111\cdots 1}_{n\text{ ...
1
vote
3answers
38 views

Help with discrete math proof

I'm having trouble with the following: $\ a_1=1$ and $a_n=1+\sum_{i=1}^{n-1} a_i$ for $n>1$ How should I go about proving the below? Any hints? $a_n = 2^{n-1}$
0
votes
1answer
13 views

Proving distance inequality between three elements in a normed linear space

For any two elements $x,y$ belonging to a normed linear space, distance between x and y is given by $\rho(x,y) = ||x-y||$ I am trying to prove the inequality $\rho(x,y) \leq \rho(x,z) + \rho(y,z)$ ...
0
votes
1answer
35 views

How to go about a “not divisible by..” proof

I need to show the following proof: For any integer x, x^2 + 4 is not divisible by 3. I was trying proof by contraposition, but I do not believe that is the most efficient way to go about this. ...
5
votes
2answers
72 views

How to decide which moduli to check when solving a “polynomial” congruence?

Consider the following problem: Find all integer solutions to $y^2 = x^5 - 4$. The solution goes something like – check modulo 11, where $x^5 \equiv 0, \pm 1$, and then check cases to arrive at ...
-3
votes
3answers
42 views

Stirling proof guidance [closed]

I would like some guidances: Prove that $$\lim_{n\to\infty}\frac{2^nn!}{n^n} = 0$$ Any help is greatly appreciate.
5
votes
4answers
80 views

Prove that if $A$ is both open and closed, $A=\mathbb R$. [duplicate]

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
1
vote
2answers
64 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
3
votes
1answer
43 views

If a sequence $f(x_n)$ goes to its minimum, will $x_n$ go to the point at which $f$ achieve the minimum?

I have a continuous function $f$ that is defined on a compact set. And $f(x_0)$ is its minimum. If I have a sequence $x_n$ such that $f(x_n)\to f(x_0)$, how can I show that $x_n\to x_0$? I tried ...
0
votes
1answer
21 views

Sum-of-divisors determinant

Let $\sigma_k(n)=\sum_{d|n}d^k$ be the generalized sum-of-divisors function. Let $S_n$ be the matrix defined by $[S_n]_{ij}=\sigma_i(j)$. I read a comment somewhere that $$\det(S_n)=1!\cdot 2!\cdots ...
2
votes
2answers
41 views

prove the nomalizer $N(H)$ of the subgroup $H$ in $G$ is a group

I need some help on the following question. For an arbitrary subgroup $H$ of the group $G$, the normalizer of $H$ in $G$ is the set $N(H) = \{x \in G \mid xHx^{-1} = H\}.$ Any help??
5
votes
4answers
94 views

Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
0
votes
0answers
12 views

Prove that on N, the relation V where mV n

This is my question! Help me, please! Prove that on $\Bbb N$, the relation $\mathsf V$ is a linear order where $m\mathsf Vn$ if and only if $m$ is odd and $n$ is even, or $m$ and $n$ are even and ...
5
votes
3answers
132 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
2
votes
1answer
61 views

“Cascade induction”?

I refer to this answer. The answer is based on several simplification steps, all of them proven by induction. $S_n = 2903^n - 803^n - 464^n + 261^n$ $T_n = 2642\cdot2903^n - 542\cdot803^n - ...
4
votes
3answers
91 views

Mistake in (Baby) Do Carmo? Elementary topology of surfaces.

If you have the book, it's proposition 2 of section 5.3. If not, the proposition reads: Given any two points p and q $\in$ a regular, connected surface S, there exists a parameterized piecewise ...
2
votes
1answer
32 views

Proving Cayley formula using Kirchhoff matrix theorem?

To count the number of spanning trees of a complete graph of order $n$ one can use Kirchhoff matrix theorem and arrive at the exact answer $n^{n-2}$. But in doing so, one should know how to evaluate ...
1
vote
2answers
29 views

Show $\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$

I have been trying to get my head around this step in a proof, but havn't been able to, Question: Show $$\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$$ ...
2
votes
1answer
49 views

Proof By Induction - $n^2 = \sum_{i=1} ^{n} (2i-1)$ for all $n\geq 1$ [duplicate]

Using Proof By Induction I am trying to prove the following: $n^2 = \sum_{i=1} ^{n} (2i-1) $ for all $n\geq 1$ Here is my solutions so Far: Base Case: $n=1, LHS: 2(1)-1 = 1, RHS = 1^2 = 1, True$ ...
1
vote
4answers
53 views

Proving Two Sets are Equal - Infinite Sets - Example

Let $$A = \{x | x = 2n+1, n\in\mathbb{Z}\}$$ and $$B = \{x | x = 2m-21, m\in\mathbb{Z}\}.$$ I am trying to prove $A =B.$ I understand that I need to prove $A\subseteq B$ and $B\subseteq A$; But my ...
0
votes
1answer
18 views

Prove that $\int_{\mathbb{R}}x^me^{2ax}e^{-x^2/2}=e^{2a^2}\int_{\mathbb{R}}(x+2a)^me^{-x^2/2}$

As the title indicates, I've been trying for quite some time now to prove that $$\int_{\mathbb{R}}x^me^{2ax}e^{-x^2/2}=e^{2a^2}\int_{\mathbb{R}}(x+2a)^me^{-x^2/2}$$ $\forall m \in \mathbb{N}, \forall ...
3
votes
2answers
89 views

show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. ...
2
votes
1answer
32 views

Epsilon and Delta proof of $\lim_{x\to0} \frac{2-\sqrt{4-x}}{ x}$

I need to prove $\lim_{x\to0} \frac{2-\sqrt{4-x}}{ x}$ I first found the limit to be $\frac{1}{4}$ by using l'hopital's rule. By definition i need to find a $\delta > 0$ for every $\epsilon >0$ ...
0
votes
0answers
15 views

Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
6
votes
3answers
169 views

Where does the proof of $\sqrt 2$ is irrational break down when trying to prove the same for $\sqrt 4$?

I'm trying find where the common proof by contradiction that $\sqrt 2$ is irrational breaks down when trying to prove $\sqrt 4$ is irrational. Assume $\left(\frac pq\right)^2=4$ and $\gcd(p,q)=1$. I ...
0
votes
3answers
47 views

Suppose $A$ is a subset of $B$ and $B$ is a subset of $C$ and $A$ is equinumerous with $C$. Prove $B$ is equinumerous with $C$.

Definition I use: $A \sim B$ means $A$ is equinumerous with $B$ which means there is a $f\colon A \rightarrow B$ that is a bijection. My goal is to prove the following, Suppose $A \subseteq ...
0
votes
1answer
38 views

Prove that $\prec$ is irreflexive and transitive

Note: Definitions I use (Velleman's How To Prove It) If $A$ and $B$ are sets, then we will say that $B$ dominates $A$, and write $A \precsim B$, if there is a function $f: A \rightarrow B$ ...
0
votes
1answer
40 views

Are $x$ and $y$ divisible by $n$, if so how do I prove it?

If $y$, $x$,are natural numbers, and $n$ is a prime number, $y = x + n$, $y>x>n$, and $y$ and $x$ are not coprime, is it true that $n$ is a divisor of both $x$ and $y$? If so could you please ...
11
votes
1answer
183 views

A proof involving the Euler phi function

Problem: Let $\varphi$ be the Euler phi function, where for any $n \in \mathbb{Z^+}$, $\varphi(n)$ is the number of positive integers less than $n$ that are relatively prime with $n$. ...
-2
votes
0answers
24 views

To prove ($A\cup B$) $\cap C$ = $(A \cup C) \cap (B \cup C)$ [duplicate]

I have never done rigorous et theory before .How do i prove this and generalise for $A_{i}$ ,i belonging to I ($A\cup B$) $\cap C$ = $(A \cup C) \cap (B \cup C)$ Hints ? Thanks
1
vote
1answer
225 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
0
votes
3answers
31 views

Proving $T(n) = 1 + \sum_{j=0}^{n-1} T(j)$, $T(0)=1$ implies $T(n)=2^n$

I feel that this is a fundamental question. $$ T(n) = 1 + \sum_{j=0}^{n-1} T(j). $$ Given $$ T(0) = 1. $$ Show $$ T(n) = 2^n. $$ If I substitute values, I can see that the series goes like 1, 2, 4, ...
0
votes
1answer
39 views

What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
4
votes
2answers
64 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
3
votes
2answers
59 views

Help in proving a tautology

I am having real trouble deriving this tautology: $\forall(x) ((x=a) \lor (x\neq a))$ It is easy to solve this by assuming the negation, unpack the negation with DeMorgan's Law, and derive from ...