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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1answer
83 views

Proof: Lemma 5.6.2 - Elements of Real Analysis (C.G. Denlinger)

I reading the book "Elements of Real Analysis, C.G.Denlinger".. and I need the proof of Lemma 5.6.2: Thanks in advance!
0
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2answers
25 views

$S=\{(x,y,z) \in \mathbb{R}^3: x^2+y^2=z^2, z \geq 0\}$ is not regular surface.

Suppose $S$ is a regular surface. There exists coordinate function $\textbf x:U \to S \cap V,$ for some open $U \subseteq \mathbb{R}^2$ and some open $V \subseteq \mathbb{R}^3$. WLOG, let $(0,0,0) ...
5
votes
1answer
93 views

Calculate the quotient groups and classify $\mathbb{Z^3}/(1, 1, 1)$ - Fraleigh p. 151 15.8

I tried to start like this question. I fill in the sizes first. $\mathbb{Z}$ is an infinite group hence $\mathbb{Z^3}$ is too. $\left|\dfrac{\mathbb{Z^3}}{<(1, 1, 1)>}\right| = ...
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0answers
13 views

Can we prove that this tabular algorithm works correctly?

Finding an answer to the following question is very important, because it will help prove an algorithm works correctly. It is also extremely hard to explain, so I'm hoping that someone will help me ...
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1answer
30 views

$\exists L \in \Bbb{R}(\forall n \in \Bbb{N}(a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n} \leq L))$?

Let be $a \in \Bbb{R}^{>0}, b \in \Bbb{R}$, $\exists L \in \Bbb{R}(\forall n \in \Bbb{N}(a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n}\leq L ))$? I thought: if $a=1 \to 1^\frac{\lfloor b \cdot ...
0
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0answers
48 views

I need help prooving a theorem in OEIS A224914

I've tried to solve this, but can't seem to get anywhere. Full description is in the pdf. http://blogoff.simonjensen.com/#post18 http://www.simonjensen.com/pdf/The_answer_is_47.pdf ...
0
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1answer
80 views

Proof: If m is an integer not equal to 0, then m is not divisible by 0. [closed]

Any help would be appreciated. If $m$ is an integer not equal to $0$, then $m$ is not divisible by $0$
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1answer
25 views

Let q be a prime and k be a integer greater than 1. Show that if x is an integer such that x^2=x(modq^k) then x=0(modq^k) or x=1(modq^k)

Apologies, the equal signs should be triple horizontals. Considering the or condition in this statement, would I be proving both could be true or two serperate cases? Also, is there some sort of ...
3
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2answers
117 views

If $a > 1$ then $a^2 > a$

I need to prove that if $a > 1$ then $a^2 > a$ At first glance it seems that this is true, if letting $a = 2$ for example than, $$2 > 1$$ and $$2^2 > 2$$ What I tried to do is work with ...
0
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1answer
33 views

Finite and Infinite Cardinality Representation

Let X be a set. Show that the cardinality of the set of finite sequences with elements from X has cardinlity $\aleph_o$ if X is finite and cardinality $|X|$ if X is infinite. I was given the hint ...
2
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2answers
119 views

Product of $n$ consecutive positive integer is not a $n$th power?

If $n>2$ and $k$ is positive integer, then there is no positive integer $m$ satisfy that $$k(k+1)\cdots (k+n-1)=m^n\, ?$$ I tried to prove this problem, but I don't know how to prove it. I know ...
1
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0answers
55 views

Replacement of sentence symbols in a well-formed formula

Suppose $\theta$ is a tautology and $A,B$ are sentence symbols occurring in $\theta$ and $\psi$ is a well formed formula obtained by replacing $B$ with $A.$ Is $\psi$ is a tautology? My proof: We ...
2
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2answers
78 views

Restate a logical claim using logical symbols

Proposition: Strictly between any two distinct rational numbers lies another rational number. How may I present this statement using logical symbols? My answer: $\forall x, y \in {\mathbb{Q}}. ...
-1
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2answers
69 views

Discrete Mathematics Proof Question

Prove or disprove that there are infinitely many $x, y, z \in \mathbb N$ such that $$\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{z^2}$$ Currently, I tried to substitute $x, y,$ and $z$ with $2n$ and $n$ ...
2
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2answers
76 views

Discrete mathematics proof relating to Fermat's Theorem

Assuming the Fermat Theorem, show that there is no natural number $x$, $y$, and $z$ and $n\geq3$ such that $$\frac{1}{x^n} + \frac{1}{y^n} = \frac{1}{z^n}. $$ So far I think proof by contradiction ...
-1
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4answers
52 views

Help with a proof in discrete math.

I have been trying to figure out this problems to no avail. Problem $1$: Show that there are infinitely many natural numbers $x,y,z$ such that $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}.$$ Thank ...
0
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2answers
87 views

Cardinality of Integers, Positive Integers, and Rational Numbers all equal $\aleph_0$

Prove that $|\mathbb{Z}|=|\mathbb{Z}^+|=|\mathbb{Q}|=\aleph_0$ I am to use cardinal addition and multiplication to reduce this to finding an injection $\mathbb{Q}^+ \to ...
1
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1answer
39 views

Axiom of regularity and ordinal ranks

I am trying to prove that the following two statements are equivalent: Axiom of regularity $\forall x \exists \alpha (\alpha $ is an ordinal and $ x \in V_\alpha)$ I believe I understand how to ...
3
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2answers
145 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
0
votes
1answer
84 views

Prove or disprove the following statement. $7 \ | \ (x^3 + x^2 + x + 2)$, where $x$ is an odd integer

We're learning about modulus and division (Discrete mathematics and proofs course). I'm not exactly sure how to tackle this sort of problem, is there some sort of property of cubic functions ...
0
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4answers
77 views

$(A_1\rightarrow\wedge A_2)$ is not a well-formed formula

Let $A_1,A_2$ be sentence symbols. Could anyone advise me how to prove $(A_1\rightarrow\wedge A_2)$ is not a well-formed formula? Thank you.
6
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2answers
87 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
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2answers
105 views

Two parallelograms are equal in area.

I tried this question by constructing a line $PD$ therefore forming two triangles $ADP$ and QDP but couldn't establish the congruency relation between the triangles. My approach was that if I have ...
2
votes
0answers
43 views

Need assistance: proof of differentiability & directional derivate of $x^2\cdot y\cdot (x^2+y^2)^{-1}$

I really need some help/guidance with the following tasks: Given is the function $x^2\cdot y\cdot (x^2+y^2)^{-1}$ for $(x,y) \neq (0,0)$ with $f(0,0)=0$. 1) Is this function partial differentiable? ...
0
votes
1answer
37 views

Proving the number of subgraphs of $G$ isomorphic to $F$

Let $F$ and $G$ be graphs. Let $sub(F, G)$ denotes the number of subgraphs of $G$ that are isomorphic to $F$, let $inj(F, G)$ denote the number of injective homomorphisms from $F$ to $G$ and let ...
2
votes
5answers
433 views

Is it acceptable to solve hypothetical statements in Linear Algebra using actual numbers?

I'm taking a Linear Algebra course this semester where we must prove/disprove hypothetical statements. So I'm wondering, is it alright to show that certain theorems hold or not using examples with ...
0
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3answers
49 views

Can't show that equations are the same

This is the problem i cant solve: Show that: $$ \frac {\tan^2(x-1)}{(\sin x+\cos x)} = \frac {(\sin x-\cos x)}{\cos^2x} $$ I can't get any longer than: Left-hand side: $$ \frac{ {\sin^2(x-1) ...
1
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2answers
169 views

Continuity of $f(x)=x^p$ when $p$ is a real number and $x\in (0,\infty)$

Here is my final answer. Definition Let $x>0$ be a real, and $\alpha$ be a real number. We define the quantity $x^{\alpha}$, by the formula $\text{lim}_{n\rightarrow\infty} x^{q_n}$ where $(q_n)$ ...
4
votes
1answer
106 views

holomorphic function with bounded real part on punctured neighborhood $\dot{D}_{\epsilon}(z_0)$

I've seen here that a holomorphic function with bounded imaginary part on a punctured neighborhood of $0$ has a removable singularity at $0$. I just wanted to know if this result could be also ...
6
votes
2answers
189 views

Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
3
votes
1answer
107 views

Tricks, Multiply across Subset - Left Coset Multiplication iff Normal - Fraleigh p. 138 Theorem 14.4

Left coset multiplication is well defined by $(aH)(bH) = (ab)H \iff H \triangle G$. Given $H\leq G$, we wish to define a group structure on $G/H$ under suitable conditions. The natural way to do ...
0
votes
1answer
31 views

Need help proving an equality

I'm looking for help proving this equality: $\forall m:m \times 0 = 0 = 0 \times m$ Any help would be greatly appreciated. Thank you very much.
1
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2answers
68 views

Proving a summation involving binomial coefficients.

I need to prove the following inductively: (http://upload.wikimedia.org/math/9/e/5/9e57871ba17c1ad48e01beb7e1bb3bb9.png) $$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$$ And for the life of me I can't ...
2
votes
1answer
68 views

Initial Segment Order Isomorphic to the Ordinal Numbers

Prove that every well-ordered proper class has an initial segment order isomorphic to the ordinal numbers, ON. I have a plan to prove this but it uses a recursive definition and induction which I do ...
1
vote
4answers
92 views

Why is $\langle \mathbb{Z}_4, + \rangle$ not isomorphic to $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$?

I'm having some trouble here, specifically with the idea of $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$ as a group. Can anyone help me out with some explanations? Moreover, I generally ...
4
votes
1answer
264 views

Problems with fake proofs of limit of sequences

I can hardly imagine an easier example of the fact that my understanding of the topic is more than rusty. I will divide the question in two parts to make the reading easier: 1) Background; 2) ...
1
vote
1answer
51 views

Question about equlaity of two language, simple but tricky.

I found the following question tricky: If $A$ is a language, when will $A^*=A^+$? By definition, $$A^* = \bigcup^{\infty}_{i=0}A^i = A^0 \cup A^1 \cup A^2 \cup \cdots$$ $$A^+ = ...
1
vote
2answers
82 views

In triangle $ABC$ prove that $AB = 2BC$

In solving this proof I am NOT permitted to use any numerically related givens (i.e., the sum of all angles in a triangle is 180 degrees or in a right triangle side Asquared + side Bsquared = side ...
1
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1answer
27 views

Proof of Exponential function $a^n$

How can I prove that $$\sum_{n=N}^M a^n = \frac{a^M - a^{N+1}}{1-a}$$ for $a$ not equal to $1$ and $$\sum_{n=N}^M a^n = M-N+1$$ for $a = 1$
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2answers
20 views

A question about operations on languages.

I come across this problem on a book. It states that: for languages A and B, $(A\cup B)^* = (A^*B^*)^*$. I know that the definition of star closure is $\left(\bigcup^{\infty}_{i=1}\right)A^i$. But so ...
2
votes
5answers
385 views

proving zeros of a polynomial are not real

I'm working on a optimization problem and need to show that \begin{equation} \frac{1}{2}x^4 - x^3 -x + 100 = 0 \end{equation} has no real solution in order to prove certain properties about the ...
0
votes
1answer
28 views

When can the length of a line be equal to a circular function?

So, I'm having a bit of trouble trying to grasp this concept. I understand that a circular function like cosine is a ratio of two sides of a triangle in reference to an angle, however, one of my ...
2
votes
0answers
136 views

Stereographic projection from sphere to $\mathbb{R}^2$

This question is from my tutorial problem set: One way to define a system of coordinates for the sphere $S^2$ given by $x^2+y^2+(z-1)^2=1$ is to consider the stereographic projection $\pi:S^2-\{N\} ...
1
vote
1answer
70 views

What are the typical approaches to showing that some function sequence does not converge uniformly?

The following problem is from Munkres's Topology (Exercise 6 of Section 21 "The Metric Topology (continued)", 2nd edition). Exercise: Define $f_n : [0,1] \to \mathbb{R}$ by the equation $f_n(x) = ...
0
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3answers
163 views

Using rules of inference (Leibniz) to prove theorems.

Leibniz: If $A \equiv B$ is a theorem, then so is $C[p:= A] \equiv C[p:= B]$. Note: p is "fresh" means p doesn't occur in $A, B, C$. I am trying to understand how to use Leibniz rule of inference for ...
0
votes
1answer
48 views

Critical Points and Gradients/Derivatives

Plot the function $f(x)= 3+\cos(3x)-0.5\sin(5x)+0.2\cos\left(10x-\left(\frac{\pi}{4}\right)\right)$. Estimate how many critical points are on the interval $[0,2\pi]$. Consider $\mathbb{R}^{20} \to ...
1
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4answers
93 views

Prove or Disprove the existence of a basis

I'm asked to prove or disprove the existence of a basis $(p_0,p_1,p_2,p_3)$ of $F(t)(3)$ (Polynomials of degree at most 3) such that each of the polynomials $p_0,p_1,p_2,p_3$ satisfies the equation ...
4
votes
0answers
74 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
0
votes
1answer
40 views

(Geometry) Proof type questions

Can someone please explain to me the given question and proof? otherwise I might just have to end up dropping my maths course because unfortunately I'm not understanding anything from my teacher. ...
0
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0answers
28 views

Validity of a proof by induction

By intuition, I would say that if L1 is a subset of L and that L is regular, then L1 is also regular, because L1 has less states than L2 and therefore there must be an automata for L1 too. However, ...