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Questions tagged [alternative-proof]

If you already have a proof for some result but want to ask for a different proof (using different methods).

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Proof for upper bound of function using an approximation through Squeeze Theorem.

I am trying to turn $n_0log_2(n_0)$ into an approximation of $nlog_2(n)$ using the following statement: $$n \geq n_0+1 $$ For a bit of context, $n$ is the number of nodes on a minimum spanning tree [...
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Let $A$ be a square matrix. Prove that $A$ ~ $I_n$ if and only if $A\vec{x} = \vec{0}$ has only the trivial solution.

I'm studying Linear Algebra for the second time, using Hoffmann & Kunze. Currently I'm trying to prove the following theorem: Theorem 7. If $A$ is an $n \times n$ matrix, then $A$ is row-...
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On exercise 14H from Willard's 'General Topology' book

$ \newcommand{\R}{\mathop{\mathbf R}} \newcommand{\FN}{\mathop{\mathfrak N}} $ The exercise is in p. 99 of the book. It says the following: Let $X$ be a topological space, and let $B(X,\R)$ denote ...
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product= $\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\left(\frac{11^{11}3^3}{13^{13}}\right)^{1/20}\sqrt{\frac{3}{7^{7/6}\pi}\sqrt{\frac2\pi}}$

$\mathrm G$ is Catalan's constant. I recently found the product $$ \alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
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76 views

If $f$ is continuous on $[a,b]$ then is bounded above in $[a,b]$ [closed]

If $∀ϵ>0:∃δ>0:|x−a|δ<⟹|f(x)−F(a)|<E$ this implies continuity, we always say that for all $E$ we can find a delta. But for all delta we can find a Epsilon right? (this seems okay for me,...
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Show $\phi(n)$ = $n \Pi_{p \vert n} (1 - \frac{1}{p})$

(Here $\phi$ is the Euler Totient Function, $p$ all the primes dividing $n$) So I wanted to know is the best way to prove this using induction on $n$? or to suppose cases where $n$ is prime or ...
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1answer
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Showing every connected regular space having more than one point is uncountable without using proof by contradiction

The common proof goes like this: Suppose $X$ is countable, then it must be Lindelöf. A regular Lindelöf space is normal (akin to the proof that a regular and second-countable space is normal). ...
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How did I solve this (triply logarithmic) equation?

In an optimiation problem I came across the following daunting equation: $$ \log \left(\frac{1-t_2}{1-y}\right) \log \left(\frac{(1-x) (t_1-t_2)}{(1-t_1) (x-y)}\right) \log \left(\frac{t_2 x}{t_1 y}\...
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Picture proof that the area of a right triangle is $xy$

I stumbled on the following result by accident: Let $A, B, C$ be the vertices of a right triangle, with opposite side lengths $a, b, c$ respectively, where $\angle C = 90^\circ$ and $a^2 + b^2 = c^...
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Where is the inclusion map being used in the proof of Corollary 2.24 from Hatcher's AT? [duplicate]

Where is the inclusion map being used here? How is proposition 2.22 being used? What is wrong with the following proof: Since $B/(A\cap B) \cong (A\cup B)/A = X/A$, then $\tilde H_n(B/(A\cap B) \...
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Prove $(\lnot p \to p) \to p$ in 19 steps

While revisiting "Mathematical Logic", I noticed a quiz that I had never worked out. It requires to prove $$\{\} \vdash (\lnot p \to p) \to p$$ in 19 steps. To be clear, each "step" should be in ...
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Elements and cyclic subgroups of order $15$ in $\Bbb Z_{30}\times \Bbb Z_{20}.$

This is Exercise 8.22 of Gallian's, "Contemporary Abstract Algebra". Please use only methods from this book prior to the exercise. This is an alternative-proof question. Find the number of ...
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Proof of exponential law using limit definition of exponential function?

For fun, I tried to prove the well-known exponential property $e^{a+b} = e^a e^b$ using the limit definition of the exponential function, below. Definition. The exponential function is defined as ...
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What is $\sigma(q^k)$ in terms of $k$ if $q \equiv 5 \pmod 8$?

Denote the sum of divisors of $x \in \mathbb{N}$ by $\sigma(x)$. Here is my question: What is $\sigma(q^k)$ in terms of $k$ if $q \equiv 5 \pmod 8$? I know that if $q \equiv 1 \pmod 8$, then $$\...
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Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus.

Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus. This inequality is equivalent to $$ 2018^{1/2018}>2019^{1/2019} $$ One of my '...
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Proof: A graph that does not contain any closed walks with odd length is bipartite.

I came across this proof in the book Mathematics for CSE by E.Lehman and F.T. Leighton(2010 version). They prove that a graph with no walks of odd length is bipartite. Here is the proof: 4 IMPLIES ...
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Finding out the remainder of $\frac{11^\text{10}-1}{100}$ using modulus [duplicate]

If $11^\text{10}-1$ is divided by $100$, then solve for '$x$' of the below term $$11^\text{10}-1 = x \pmod{100}$$ Whatever I tried: $11^\text{2} \equiv 21 \pmod{100}$.....(1) $(11^\text{2})^\text{2}...
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If $q^k n^2$ is an odd perfect number with special prime $q$, then $n^2 - q^k$ is not a square.

Problem Statement Prove the following proposition. If $q^k n^2$ is an odd perfect number with special prime $q$, then $n^2 - q^k$ is not a square. Motivation Let $q^k n^2$ be an odd perfect ...
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A more natural solution to finding the general terms of a recurrence relation in $2$ variables

A high school contest math problem in a problem book: Find the general terms of $$a_{1}=a,\quad b_{1}=b,\quad a_{ n + 1 }=\frac { 2 a _ { n } b _ { n } } { a _ { n } + b _ { n } },\quad b_{ n ...
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Why is $z = 1 + 1 + \dots + 1$ for all positive $z \in \mathbb{Z}$

How to prove that for all $z \in \mathbb{Z}$ with $z > 0$ there is an $n \in \mathbb{N}$ with $n > 0$ such that $z = \underbrace{1 + 1 + \dots + 1}_{n \text{ times}}$, or as a logical formula: $...
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For distinct primes $p$ and $q$, does any group of order $p^2q$ have a subgroup of order $p$ (without using Sylow or Cauchy)?

The Details: I'm reading "Contemporary Abstract Algebra (Eighth Edition)," by Gallian. This is based on Exercise 7.40 of the "Cosets and Lagrange's Theorem" section ibid. Here it is for convenience: ...
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1answer
24 views

Riesz representation theorem: show isometry

Let $X$ be a Hilbert space and $J:X\rightarrow X',J(X):=(\cdot,x)$ where $X'$ is the dual space of $X$. I have to show that $\|J(x)\|_{\sup}=\|x\|$. ''$\leq$'' is clear by the Cauchy-Schwarz ...
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A single set of moves $S$ that, if repeated, solves the Rubik's cube from any state

I am looking for a proof verification. I often find these concepts simple, but struggle to communicate them clearly. Communication in mathematics is very important to me: Examples could be: Any ...
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2answers
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Hilbert Space: Orthogonal projection is linear

Let $X$ be a Hilbert space and $A\subset X$ a closed subspace show that the orthognal projection $P:X\rightarrow A$ is linear. Now I know that $x-P(x)\in A^{\perp}$. The lecture notes go on by saying ...
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Hilbert space $x-P(x)\in A^{\perp}\Rightarrow \|x-P(x)\|=d(x,A)$

Let $X$ be a Hilbert space and $A\subset X$ a closed subspace. Let $P:X\rightarrow A$ be a mapping for wich $x-P(x)\in A^{\perp}$ holds. Show that $\|x-P(x)\|=d(x,A)$ $\hspace{2cm}$ $(1)$ where $...
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Multiple proofs of $\sum_{d|n}{\phi(d)}=n$ [duplicate]

I am looking for multiple proofs of that statement: here $\phi(n)$ denotes the Euler’s totient $$\sum_{d|n}{\phi(d)}=n$$ Here’s one: By unique factorisation theorem: $n=\prod_{k=1}^{m}{p_k^{\...
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1answer
39 views

Show that $a_{k-1}a_{k+1}\le a_k^2$ where $P(x) = \sum_{k=0}^n a_k X^k$ is a real polynomial with real roots, and $0<k<n$

the question is in the title. I know that this is a direct consequence of Newton's inequalities but I'm looking for a proof without using it. A hint was given to solve it : Show that $$ P'^2 - P'...
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Proof of separability of polynomials without derivatives

Is there a known proof without differentiating that proves that all irreducible polynomials over $\mathbb{Q}$ are separable? (Or even better, for all fields of characteristic $0$.) EDIT: As people ...
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Direct proof of Archimedean Property (not by contradiction)

I looked at the proof of Archimedean Property in several places and, in all of them, it is proven using the following structure (proof by contradiction), without much variation: If $\space x \in \...
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If |x + y| > |x - y|, then how to arrive at $xy > 0$?

When $\lvert x + y\rvert > \lvert x - y\rvert$, I am aware that we can square both sides to find that $xy > 0$. $x^2 + 2xy + y^2 > x^2 - 2xy + y^2$ $4xy > 0$ $xy > 0$ However, I'm ...
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Alternate proof for a problem using logic

I applied Modus Tollens and De Morgan's laws to the following property of primes: If $p$ is a prime and $p | ab$ for $a, b \in \mathbb{Z}$, then either $p|a$ or $p|b$. Modus Tollens: If not ($p|a$)...
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Proving $(ab)^{-1} = a^{-1}b^{-1}$, if $a,b\ne 0$

Having only these axioms: add associativity. add identity. add inverse. add commutative. mul associativity. mul identity. mul inverse. mul commutative. distributive. Prove that $(ab)^{-1} = a^{-1}b^{...
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Validity of Using Induction to Show Union of an Infinite Ascending Chain of Subgroups is a Subgroup

Can this be done by induction instead of just proving the subgroup criterion? I can prove using the essentials tools of group theory, but looking at the problem, I was wondering if we can simply use ...
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1answer
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Verification of alternative proof of $\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$

I have to show that $$\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$$ Suppose $u\in L^\infty (E) $ for measurable $E \subset \mathbb{R}^d$ having finite measure. I come up with this proof: $$\Big| \|u\|_p -...
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Surjectivity of composition of functions

Assume $F: X\rightarrow Y$ and $G: Y \rightarrow Z$ are surjective prove that $G \circ F$ is surjective. Let $z\in Z$ Set x$\in X$ to be such that $F(x)=y$ and $G(y)=z$ (these exist because F and G ...
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Better proof for $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$

Prove this $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$ My attempt Proof - by using [axiomdistributive] and [axiommulcommutative]: $$\begin{split} &(x+y)(x^2 - xy + y^2)\\ &= (x+y)x^2 - (x+y)...
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3answers
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Prove $n{2n\choose n}=(n+1){2n\choose n-1}$ and $(n+1)|{2n\choose n}$

a) Show that $n \cdot {2n\choose n}=(n+1){2n\choose n-1}$ for all $n \in \mathbb{N}$. b) Show that $(n+1)\mid{2n\choose n}$ for all $n \in \mathbb{N}$. Prove for a) $$n\cdot {2n\choose n} =n\...
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How can I find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$ efficiently with combinatorics?

To find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$, I used factorization on $(1+x+\frac{x^2}{2})$ to obtain $\frac{((x+(1+i))(x+(1-i)))}{2}$, then simplified the question to finding the ...
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2answers
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Subtracting ratios from each other to find which solution is more concentrated.

3 litres of orange concentrate were mixed with 5 litres of water to make a drink. Later, 2 litres of orange were mixed with 3 litres of water. Which mix is more concentrated? Consider the following ...
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Find at least ten different prime numbers $p$ such that $p + 6$ is also prime. [closed]

How can I find the set of prime numbers without using the direct trial and error?
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Prove that $\phi(f(X),Y)=\phi(X,f(Y))~\forall X,Y\in\mathbb R^3$ where $\phi(X,Y)=X^TAY$ and $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$

Within this AoPS thread there was the following question asked Let $\phi(X,Y)=X^TAY$ be a scalar product on $\mathbb R^3,$ and let $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$, where $$A=\begin{...
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1answer
26 views

Proving a definition and the general term for this sequence

Consider the sequence $(T_n) = 1, 4, 9, 16 ...$ In and exercise I'm trying to solve I'm asked to: A) Show that $(T_n)$ is defined by $T_n=1+3+5+7+...+2n-1$ B) Prove that the general term for $...
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2answers
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Find the length of a triangle's side

I have the following triangle and I'm supposed to find the value of x. First thought that came to mind is to use the the following tangent equation $$\tan(y)=\frac{x}{27}$$ and $$\tan(19+y) = \frac{...
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Every CW subcomplex is a strong deformation retract of some open set

I am reading Rotman Algebraic topology for CW complexes. On page 210, the author has given a proof of the following theorem :--- Let $(X,E)$ be a CW complex and $(Y,E')$ be a subcomplex and let $M$ ...
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34 views

Quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$

I'm looking for a quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$ for some nonnegative integers $e_p$. I've argued as follows, but I'm hoping for a '...
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2answers
79 views

Rigorous or not?

I want to prove $$T(n,k)=\frac{n}{\left\lfloor\frac{n+k+1}{2}\right\rfloor}(T(n-1,k)+T(n-1,k-1))=\binom{n}{k}\binom{n-k}{\left\lfloor\frac{n-k}{2}\right\rfloor}$$ First we know only that $$T(n,k)=0, n&...
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2answers
59 views

Find all $P\in \mathbb{C}[X]$ such that $P(\mathbb{Q})\subset \mathbb{Q}$ [duplicate]

I came up with a proof for this question, rather simple, that I didn't find here. And I wondered if my proof was correct, and if anyone had a different proof ? Thanks for you time. I will show that ...
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0answers
38 views

Proving $(U \cap V) \cap u(U \cap v) = (u \cap V)(U \cap v)$

Let $U, V$ be subgroups of a group. Let $u \trianglelefteq U, v \trianglelefteq V$. I proved like this. Then by applying modular law, $$\mathrm{(LHS)} = (U \cap V) \cap u(U \cap v) = U \cap V \cap uv \...
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0answers
75 views

Ordering integer multiplets

I have proved the following result in arXiv:1005.3371v8 Lemma 3.31: Let $n$ be a natural number greater than or equal to 2. There exists a bijection $f$ from $\mathbb{N}$ onto $\mathbb{Z}^n$ so that $...
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107 views

Is there a proof of quadratic reciprocity using $p$-adic numbers?

I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. ...