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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0
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1answer
21 views

seat every club member np problem

A University has n clubs, the largest of which contains m members (students can be members of multiple clubs). The President of the University wishes to hold a dinner in honor of such student ...
2
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5answers
126 views

Proving that if $a^2+b^2=c^2$, then $a+b\ge c$.

Hello, I'm trying to prove this statement. Let a,b & c be three positive real numbers and if $a^2+b^2=c^2$ then $a+b\ge c$ Any help, please?
3
votes
2answers
145 views

$x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$ [duplicate]

Problem Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $\vert xy \vert = n$. Solution We have $x^2 = ...
0
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1answer
67 views

How do i prove $\text{F}_\text{n+1}^2 - F_n\text{F}_\text{n+2} = (-1)^n$ by induction? [duplicate]

$F_n$ refers to the $n$ term of the Fibonacci Sequence. I think i'm suppose to prove this by induction. I already have the base case. I am at: $\text{F}_\text{k+1}^2 - F_k\text{F}_\text{k+2} + ...
3
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4answers
1k views

Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.

Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
2
votes
3answers
405 views

Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.

Is there a better (or other) way(s) to prove the following statement? Also, the same argument works for multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{Q}-\{0\}$, right? Problem Prove that ...
13
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2answers
495 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
6
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3answers
168 views

Intuition and Tricks - Crafty Short Proof - Generators, Order of a Cyclic Group - Fraleigh p. 64 Theorem 6.14

This stronger result and easier proof is based on p. 58. Hence it isn't a duplicate of this. Theorem 206 and 207. Let $G$ be a group, $k \in \mathbb{N}$ and $a \in G$ such that $|a| = n$. Then: 206. ...
2
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1answer
72 views

Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements

I would like to know if my proof below is correct. Problem Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements. Solution If $\mathbb{F}$ is a ...
4
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3answers
130 views

Proving that $L \le M$ as limits of $f$ and $g$ when $f(x) \le g(x)$

While doing some tasks for my next calculus course, I ran across this task: "Let $a < b <c$, and assume that $f(x) \le g(x)$ for all $x \in [a, c]$. If $\lim_{x \to b}f(x) = L$ and $\lim_{x \to ...
2
votes
2answers
70 views

If $\omega$ is closed on $\mathbb R ^2 - 0$ and $\text d \omega =0$, then $\omega = \text d g+ \lambda \text d \theta$.

I'm trying to solve problem 4-30 from “Calculus on manifolds”, which is the one in the title, where $$\text d \theta = -\dfrac{y}{x^2+y^2}\text d x+\dfrac{x}{x^2+y^2}\text d y.$$ I think I'm on the ...
5
votes
3answers
190 views

Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
2
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0answers
190 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
1
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3answers
108 views

countable or not countable

Good evening everyone; Can you tell me whether are they countable or not ? $$ 2^\mathbb{R}\\ 2^\mathbb{Z}\\ 2^\mathbb{N} $$ where $\mathbb{R}$ is the set of real numbers. $\mathbb{Z}$ is set of ...
0
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1answer
196 views

Mathematical derivation of successive parabolic interpolation

I'm studying for an exam, and in some past papers they ask to provide 'a mathematical derivation of the iteration scheme'. Now, I understand successive parabolic interpolation, I can do it (or I ...
1
vote
0answers
70 views

Prove that $\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3$ [duplicate]

Prove that $$\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3$$ where $a$ and $b$ and $c$ are positive real numbers. How to do it? Thank you
1
vote
1answer
85 views

If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some $k\in\mathbb N$.

I would like to know if my proof below is correct. Problem If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some integer $k$. ...
3
votes
1answer
71 views

Elementary number theory problem

Let $X = \{n \in \mathbb{N}: 6 \times n\,\, \text{does not consist of} \ 0,1,2,3 \, \text{or} \ 4\}.$ For eg, $93 \in X$ because $6 \times 93=558.$ Could anyone advise me how to prove there ...
0
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2answers
256 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
5
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1answer
516 views

Tricks to Prove Union of Two Subgroups iff One is Contained in the Other - Fraleigh p. 54 - based on Exercise 5.4.5

Not a duplicate because I'm asking about tricks and the blueprint for the proof based on this. Let $H, K \le G$. Prove $H \cup K$ is a subgroup $\iff H \subseteq K$ or $K \subseteq H$. Backward ...
4
votes
2answers
400 views

Intuition - An inverse of a generator is a generator and powers of generators - Fraleigh p. 58 5.46

(1). How do you envisage or envision a Cyclic Group with only one generator can have at most 2 elements? Solution is based on this. The integers have two generators, 1 and -1. Which should give you ...
0
votes
2answers
67 views

continuous map from indiscrete space

Show that a continuous map from indiscrete space X to a $T_0$ space must be constant function. I was trying to prove it by contradiction. If the map f is not constant then there is two points x, y ...
4
votes
4answers
534 views

If we accept a false statement, can we prove anything? [duplicate]

I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$) to ...
2
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2answers
75 views

combinatorics proof with i - Peter J. Cameron book

I'm working through Peter J. Cameron's combinatorics book and I'm having trouble understanding one of his proofs. In proposition 3.3.3, he states: "If n is a multiple of $8$, then the number of sets ...
0
votes
1answer
69 views

Finishing proof of identity $\sum_{k=b}^{n} \binom{n}{k} \binom{k}{b} = 2^{n-b} \binom{n}{b}$

The identity $$ \sum_{k=b}^{n} \binom{n}{k} \binom{k}{b} = 2^{n-b} \binom{n}{b}\ $$ is one of a few combinatorial identities I having been trying to prove, and it has taken me way too long. I am ...
0
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5answers
171 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
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2answers
132 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
330 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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4answers
91 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 ...
18
votes
2answers
955 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 ...
13
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4answers
466 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
90 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
96 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
141 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
vote
2answers
87 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
votes
1answer
73 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
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1answer
19 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
196 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
vote
3answers
171 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
238 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
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2answers
54 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
88 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 ...
3
votes
9answers
191 views
0
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3answers
46 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
vote
1answer
145 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
vote
3answers
142 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
103 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
1answer
35 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
505 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.