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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5answers
167 views

Questions about Divisibility of $2^n$ by $3$

Why is it that $\forall n \in N$, $2^n$ is not divisible by $3$? I can prove it easily by induction, but I don't understand the intuition of why this is true. Could anyone please supply the ...
6
votes
2answers
122 views

Intuition/How to determine if onto or 1-1, given composition of g and f is identity. [GChart 3e P239 9.72]

9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions. Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? $1.$ $f$ ...
5
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2answers
308 views

Proof Strategy - Tricky Question involving distinct, non-zero eigenvalues of $A^{2}$ - 2011 8C

Remember that we've already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors ...
3
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3answers
73 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
16
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2answers
797 views

Integral $\int_1^\infty\dfrac{dx}{1+2^x+3^x}$

Can the integral $$\int_1^\infty\dfrac{dx}{1+2^x+3^x}$$ be given in closed form? This question arises naturally when I considered doing integrals. What makes an integral hard? Well, the integrand, of ...
12
votes
4answers
331 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
0
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2answers
73 views

Verification of Proof of a Bijection from A to B

Problem: For $ a,b \in \textbf{R}$ with $ a < b$, prove an explicit bijection of $ A = \{ x : a < x < b \} $ onto $ B = \{ y : 0 < y < 1\} $. My attempt: We consider $ f(x) = ...
4
votes
1answer
83 views

Give the definition of a binary operation for these to be isomorphisms - Fraleigh p. 34 3.18

(1.) In blue: I understand $\phi^{-1}$ is a homomorphism $\iff \phi^{-1}(3a - 1) * \phi^{-1}(3b - 1) = \phi^{-1}((3a - 1) + (3b - 1)).$ But where did the $(3a - 1), (3b - 1)$ crop up from? Why not ...
4
votes
2answers
127 views

$\mathbb{C}[x,y]/(x^2+y^2+1)$ is an integral domain.

I stuck in the following question. Prove that $ \mathbb{C}[x,y]/\langle x^2+y^2+1 \rangle $ is an integral domain, using the following: Let $\mathbb{F}$ be a field, $c \in \mathbb{F} $. ...
1
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2answers
76 views

Can consecutive integers be perfect powers?

I have been wondering whether consecutive integers can ever be perfect powers.And even if they can, how many consecutive integers at most can be perfect powers?My intuition tells me that consecutive ...
0
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1answer
57 views

Transitive closure of binary relation with proof of equivalence

On the set X = {1,2,3,4,5,6,7,8,9}, there is binary relation Q = {(1,9),(2,5),(3,7),(4,1),(5,8),(6,2),(7,3),(8,6),(9,4)}. Make a transitive closure T of the relation Q. Decide and prove whether the ...
0
votes
1answer
18 views

Prove that there are three real numbers a and b and c so that: $(\forall x\ne -1/2)\frac{6x^2+7x-3}{2x+1}=ax+b+\frac{c}{2x+1}$

1) Prove that there are three real numbers a and b and c so that: $(\forall x\ne -1/2)\frac{6x^2+7x-3}{2x+1}=ax+b+\frac{c}{2x+1}$ [[Addition: this question 1) is related to this question 2): ...
6
votes
3answers
181 views

Is there a proof for this Fibonacci relationship?

I was looking at the decomposition of Fibonacci numbers using the definition of $F_n = F_{n-1} + F_{n-2}$, and noted the pattern in the coefficients of the terms were Fibonacci numbers. It appears to ...
0
votes
1answer
49 views

real analysis help $f(x)\ f'(x)$

Let $f(x)$ be a differentiable function such that $f ′ (x)$ is continuous. If $f(0)=0$ and $f(x)\ f ′ (x)\le 0$ for all $x\gt 0$ , prove that $f(x)=0$ for all $x\ge 0$ . attempt: I know you ...
1
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3answers
135 views

Prove If $a^3>a$ then $a^5>a$

Prove If $a^3>a$ then $a^5>a$ Here was my go at it: Assume $a^3>a$. Then $$a^3>a\Rightarrow a^3-a>0\Rightarrow a(a+1)(a-1)>0$$ Solving this inequality gives the truth set ...
1
vote
1answer
102 views

Proving that something is an affine subspace

I'm trying to prove that a given subset of a given vector space is an affine subspace. Now I'm having some trouble with the definition of an affine subspace and I'm not sure whether I have a firm ...
0
votes
2answers
44 views

Significance of low order terms in base expansion of integer square root

My head is turning into a uniform gel of random thoughts! I cannot see a proof or find a counterexample to the following: Conjecture: Let integer $x$ be expressed as $a_3 \, b^3 + a_2 \, b^2 + a_1 \, ...
1
vote
2answers
81 views

Multiplicative inverses for $Z_n$

Whilst reading I came across the strange claim that multiplicative inverses exist for only prime values of $n$ in $Z_n$. I am a little puzzled as contrary to that, I know that additive inverses exist ...
0
votes
8answers
106 views

How to derive this formula? $\displaystyle n^2=\sum_{k=1}^n(2k-1).$

Welcome sir in the content of my question. How to derive this formula? $$n^2=\sum_{k=1}^n(2k-1).$$ Thank you sir.
0
votes
3answers
36 views

Proofs regarding composition of functions

I'm having trouble approaching the following question: Is the following statement true or false, provide a proof or a counterexample. If $h: A\rightarrow B, \ g: B\rightarrow C, \ f: B\rightarrow ...
1
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1answer
119 views

Book covering introduction to mathematical proofs

I am looking for some introductory books covering mathematical proofs, axioms, propositions, proof techniques etc in general.
1
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3answers
116 views

How do I derive $(\forall x)(\forall y)(\exists z)(x = y \circ z)$ from these three group axioms and some previously established theorems?

I am currently self-studying Patrick Suppes' Introduction to Logic and I am stuck on exercise 5.2.4. I've successfully worked out proofs for Theorems 1 to 7, but I am having trouble coming up with a ...
4
votes
4answers
90 views

Prove that for vectors $v_1,…,v_n$ in $\mathbb C^n$, $\{v_1,…,v_n\}$ is a basis for $\mathbb C^n$ iff its conjugate is a basis for $\mathbb C^n$

Prove that for vectors $v_1,...,v_n$ in $\mathbb C^n$, $\{v_1,...,v_n\}$ is a basis for $\mathbb C^n$ if and only if $\{\bar v_1,..., \bar v_n\}$ is a basis for $\mathbb C^n$. I know intuitively that ...
0
votes
0answers
38 views

Why does this proof need another case?

A psuedograph (with at least two vertices) is Eulerian if and only if it is connected and every evertex is even. Proof: (-->) understood so let's move on. (<--) For the converse, suppose that G is ...
0
votes
1answer
29 views

Help with proving set operations

Assuming that we have three sets, A, B, and C, and we have the implication B - C is a subset of A complement, implying A intersect B is a subset of C. How would I prove this statement? Edit: I made a ...
0
votes
3answers
444 views

Prove if $a \equiv c \pmod n$ and $b \equiv d \pmod n$ then $ab \equiv cd \pmod n$.

Prove if $a \equiv c \pmod n$ and $b \equiv d \pmod n$ then $ab \equiv cd pmod n$. I tried to use $(a-c)(b-d) = ab-ad-cb+cd$, but it seem doesn't work.
1
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3answers
67 views

Homological problem help

Need help solving this question, In general, I read the Wikipedia page on it and I realize that it does not contain the necessary knowledge to visualize this problem. Suppose $S$ is a closed ...
2
votes
1answer
63 views

Colimits and epimorphisms.

I am working on a project, and I need to know the proof of this: Any functor which preserves all colimits preserves epimorphisms. So could you please tell me how or where I can find the proof ...
1
vote
1answer
61 views

What is the proof that $\sum \limits_{v \in V} deg(v) = 2|E|$?

My textbook gives $\sum \limits_{v \in V} deg(v) = 2|E|$ and has the proof If an edge is not a loop it gets counted twice b/c it's incident with 2 different vertices. If an edge is a loop, by ...
0
votes
1answer
81 views

how I prove a valid recursion?

Determine whether is this a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined (valid recursion), find a formula for f(n) when ...
-1
votes
4answers
80 views

Given the set of eigenvalues of a diagonalizable matrix, show that it satisfies an equation

Let $A$ be an $n \times n$ diagonalizable matrix. $A$ has only $2$ and $4$ as its eigenvalues. Show that $A^2 = 6A − 8I$. I get stuck on this question for a while. Can anyone give me a hint for this ...
0
votes
1answer
60 views

Find the number of subsets $S$ of $X$ (of any size) that satisfy the following property

Let $X=\{1,2,\dots,10\}$ define the relation $R$ on $X$ by: for all $a,b\in X$, $a\mathrel{R}b \iff ab$ is even. 1) Find the number of subsets $S$ of $X$ (of any size) that satisfy the ...
0
votes
2answers
73 views

Determining whether a relation is reflexive, symmetric, transitive.

Let $X=\{0,1,2,...,10\}.$ Define the relation $R$ on $X$ by: for all $a,b$ in $X$, $a\mathrel{R}b$ if and only if $a+b=10$ is $R$ reflexive? symmetric? transitive? $a\mathrel{R}a$ $a+a=10$ ...
0
votes
1answer
71 views

Gauss-Bonnet theorem question

I was wondering if someone here can give me a hand with the proof in the image below. This is not HW, just a brain-teaser I am working on. Prove. $2\pi \chi(M)=\sum\limits_{v_i}k(v_i)$
0
votes
0answers
22 views

Rank of a matrix equivalent statements

I am working with the rank of a matrix in a proof I am working on but I am getting a bit stuck. A have two inveritable, real $p \times p$ matrices $A,B$ such that $AB\textbf{x} = \textbf b$. Is there ...
1
vote
2answers
92 views

Formal proof involving existential quantifier

It is common sense that to derive a formula with existential quantifier is only necessary to prove that a formula is valid for any term , ie: $\Gamma$ , $\phi$ [t/x] $\vdash$ $\exists$x$\phi$. By ...
1
vote
2answers
67 views

Cantor-Schroder-Bernstein Contradiction

I need help figuring out where to start a proof that says I should use a proof by contradiction. $f\colon A\to B$ and $g\colon B\to A$ be functions and each is 1-1. Let $D$ be the range of $f$ (i.e., ...
2
votes
1answer
38 views

Analysis question about finding the limit.

Consider the sequence $(c_n)_n$ defined recursively by $c_1=1/2$ and then $c_{n+1} = \frac{3+c_n}{4}$. Show that this sequence increases to a finite limit $L$ and find $L$. I just need perhaps one or ...
8
votes
3answers
124 views

What to look for in a proof?

I am a physics undergrad, wishing to pursue a PhD in Math. I am mostly self taught in the typical math undergrad curriculum. I am looking for more input, in ways I can improve my mathematical ...
0
votes
1answer
81 views

Proof polynomial is always divisible by number

Given $f(x) \in \mathbb{Z} [x] $ a polinomyal, that evaluated in any $a \in \mathbb{N} $, results allways in a multiple of 101 or a multiple of 107 (both prime numbers). Prove then, that $f(x)$ it's ...
0
votes
1answer
71 views

Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that… [duplicate]

Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that $$\lim_{x\to 0}\bigg(\dfrac{a_1^x+a_2^x+\cdots+a_n^x}{n}\bigg)^{\frac{1}{x}}=\sqrt[n]{a_1a_2\cdots a_n}$$ Got no clue where to begin from ...
1
vote
1answer
25 views

Prove Bijection in roots of unity function

Given $k \in \mathbb{N}, G_k = \{z \in \mathbb{C} |z^k =1 \} $. Probe that if $n$ and $m$ are coprime, the function $f: G_n \times G_m \rightarrow G_{mn}, f(\alpha, \beta) =\alpha\beta$ is bijective. ...
0
votes
1answer
55 views

Cauchy convergence test of a sequence

I need to prove that the sequence $ x_n = \frac{1}{n+1} $ Converges using the Cauchy convergence test. Now I know the following $ \forall \epsilon > 0 \enspace \exists N \enspace \forall n \geq ...
1
vote
1answer
60 views

Let $x$ be a real number. To prove…

Let $x$ be a real number. Define the sequence $(x_n)_{n\ge1}$ recursively by $x_1=1$ and $x_{n+1}=x^n+nx_n$ for $n\ge1$. Prove that, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
17
votes
5answers
514 views

Proving a certain map on the closed unit disc must be the identity

Bounty expired. Will gladly re-create one if a satisfactory answer is posted in the future. Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the ...
0
votes
2answers
32 views

Countable or Not Countable

Is this countable/not countable? a)The set of all ratio of two poly at most degree 3 with integer coef. is countable. b) Set of ratio of polynomial with integer coef.
0
votes
1answer
42 views

Sum of distances of a sequence is bounded $\longrightarrow$ sequence is Cauchy

A sequence $\{c_n\}$ in a metric space $(X,d)$ $(euclidean$ $distance)$ satisfies the following condition: There exists a positive real number $R\in \mathbb {R_{+}}$ such that for all $n\in \mathbb ...
0
votes
1answer
48 views

Does this constitute sufficient proof? [duplicate]

Task I have the following function $f(x)=x^2+1$ I need to prove, according to the $\epsilon - \delta$ definition of a limit, that $f(x)$ is continuous at $x = 2$. Step 1 $\forall \epsilon > 0 ...
4
votes
1answer
83 views

Application of Mean Value Theorem and Interval

Using the mean value theorem establish the inequality $$7\frac{1}{4}<\sqrt{53}<7\frac{2}{7}$$ This is obviously a true statement but can you help me form the interval and what function I should ...
1
vote
2answers
101 views

Proving $L=\{0^n \mid \text{n is a perfect square}\}$ is not a Regular Language without the Pumping Lemma

Is this a valid way of going about proving the proposition? Let $L = \{0^n \mid \text{n is a perfect square}\}$. The regular languages are closed under concatenation. So if $x \in L, y \in L$, then ...