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
0answers
88 views

Proof for the distributivity of multiplication over addition for a Binary Field

For the standard binary field $\mathbb{F}_{2} = \{0, 1\}$. Where the operations of addition and multiplication exist, and multiplication is equivalent to logical and, and addition is equivalent to ...
0
votes
1answer
80 views

Groetzsch Graph planarity [closed]

(1) Prove that the Groetzsch Graph is not planar.
5
votes
2answers
363 views

Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
1
vote
2answers
94 views

Let $f,g \in {\mathscr R[a,b]}.$ If $\int^{b}_{a}f^2=0,$ then $\int^{b}_{a}fg=0.$

Let $f,g \in {\mathscr R[a,b]}.$ If $\int^{b}_{a}f^2=0,$ then $\int^{b}_{a}fg=0.$ I have shown that $2|\int^{b}_{a}fg|\leq t\int^{b}_{a}f^2 + \frac{1}{t}\int^{b}_{a}g^2, t>0.$ Hence how do I show ...
7
votes
3answers
171 views

If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:

If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then: $$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) ...
2
votes
1answer
88 views

Use induction on $n$ to prove that $2n+1<2^n$ for all integers $n≥3$.

Use induction on n to prove that $2n+1<2^n$ for all integers $n\geq 3$. My attempt: Let $P(n)$ be the statement $2n+1<2^n$. Base case: Prove that $P(3)$ is true. $LS = 2(3)+1=7$ and ...
4
votes
1answer
130 views

Prove that there exists a scalar potential $f( \mathbf{ x} )$ such that $\mathbf{ F } = − \nabla f$ [2012 11c]

Question: If F is an irrotational vector field (i.e. $ \nabla \times \mathbf{ F = 0 }$ everywhere), prove that there exists a scalar potential $f( \mathbf{ x} )$ such that $\mathbf{ F } = − ...
0
votes
2answers
97 views

Coset of an infinite group

I am required to describe the cosets of the following subgroups: $\left<\frac 1 2\right>$ of $\mathbb R^\times$ and $\left<1/2\right>$ of $(\mathbb R,+)$. I think I was able to define ...
1
vote
1answer
84 views

How can I end my proof about a solution of $x^2-2=0$ in $\mathbb{R}$?

In the class we have constructed the real numbers using Dedekind Cuts. Now an exercise ask me to prove that the equation $x^2-2=0$ has a solution in $\mathbb{R}$ using The Least Upper Bound Property. ...
0
votes
2answers
98 views

Planar Graphs Question

I'm having some trouble with planar graphs. Two questions I was stuck on were: Prove that each planar graph on $n \gt 3$ vertices will have a minimum of $4$ vertices of degree $5$ at most. Let's say ...
1
vote
3answers
82 views

Partition of an equivalence relation

I am having a hard time with the following problem: In F(R), let f~g iff f(x)=g(x) for all x>c where c is some fixed real number. I proved that it was a equivalence relation by the following: ...
2
votes
3answers
516 views

Floor and ceiling function proof

I have the following to prove: $$\lfloor 3x\rfloor = \lfloor x\rfloor + \left\lfloor x+\frac 13 \right\rfloor + \left\lfloor x+\frac 23 \right\rfloor $$ The definition of a floor function is: $ ...
4
votes
2answers
45 views

If $\psi:G \to H$ is a surjective homomorphism, then $|\{g \in G: \psi(g)=h_1\}| = |\{g \in G: \psi(g)=h_2\}|, \forall h_1,h_2 \in H.$

If $\psi:G \to H$ is a surjective homomorphism, then $|\{g \in G: \psi(g)=h_1\}| = |\{g \in G: \psi(g)=h_2\}|, \forall h_1,h_2 \in H.$ Could anyone advise on the proof? If $\psi$ is injective, then ...
0
votes
2answers
81 views

Using induction to prove $a_n >2^n$

For the sequence $a_n=2a_{n-1}+1$ where $a_0=1$ Show that $a_n>2^n$ using induction. Use proof by contradiction (minimum counterexample). Attempt: 1. I assume, that ...
1
vote
3answers
101 views

Proof of a trigonometric expression

Let $f(x) = (\sin \frac{πx}{7})^{-1}$. Prove that $f(3) + f(2) = f(1)$. This is another trig question, which I cannot get how to start with. Sum to product identities also did not work.
8
votes
2answers
444 views

Hard contest type trigonometry proof

Suppose that real numbers $x, y, z$ satisfy: $$\frac{\cos x + \cos y + \cos z}{\cos(x + y + z)} = \frac{\sin x + \sin y + \sin z}{\sin (x + y + z )} = p$$ Then prove that: $$\cos (x + y) + \cos (y + ...
8
votes
1answer
172 views

Intermediate field, normal closure and Galois group

Let $K/F$ be Galois with $G=Gal(K/F)$ and let $L$ be an intermediate field. Let $N\subseteq K$ be the normal closure of $L/F$. If $H=Gal(K/L)$ show that $Gal(K/N)=\cap_{\sigma\in G}\sigma ...
2
votes
2answers
120 views

How to prove that if $-1<x<0$ then $x^2 + x < 0$?

I am trying to prove an equivalence. I have already proved that: $$x^2 + x < 0 \implies -1 < x < 0 $$ using a sub-proof by cases, in which I used the fact that when $xy < 0$, $x$ and ...
0
votes
1answer
49 views

Properties of equivalence relations

Let $\sim_1$ and $\sim_2$ be distinct equivalence relations on $A$. Define $\sim_3$ by $a\sim_1 b$ and $a\sim_2 b$. Let $[x]_i$ denote the equivalence class of $x$ for $\sim_i$ ($i=1,2,3$). Prove ...
1
vote
3answers
397 views

Prove that is $A$ is skew-symmetric, then $X^TAX = 0$ for all $X = [x_1 x_2 \cdots x_n]^T$ [duplicate]

Recall that a matrix $A$ is skew-symmetric if and only if $A^T = -A$. Prove that if $A$ is skew-symmetric, then $X^TAX = 0$ for all $X = [x_1 x_2 \cdots x_n]^T$
2
votes
1answer
73 views

Where is the error? Determining why a proof is incorrect.

Why is the following proof incorrect? I have tried to find out why for a few hours... The answer is: We aren't really proving the conclusion of each case is valid for the other case. Consider the ...
5
votes
5answers
257 views

Proving $n^3$ is even iff $n$ is even

I am trying to prove the following statement: Prove $n^3$ is even iff n is even. Translated into symbols we have: $n^3$ is even $\iff$ $n$ is even Since it's a double implication, I ...
0
votes
1answer
56 views

If $T$ is injective then there exists $\alpha>0$ such that $||Tx||\geq \alpha||x||$

Is this proof correct? I'm proving that if $T$ is a linear operator whose is injective then exist $\alpha>0$ such that $$||Tx||\geq\alpha||x||$$ for all $x$. By contrapositive. Assume that for all ...
-1
votes
5answers
986 views

Proving sets A and B are countable

a) Let A and B be disjoint sets, which are both countable. Prove that $A$ U $B$ is also countable. b) Use part (a) to show that the set of all irrational real numbers is not countable. So for part a ...
0
votes
1answer
53 views

Is that proof correct? I'm proving that the set of injective Linear transformations in open.

Let $T$ an inyective linear transfomation from $\mathbb{R}^n$ to $\mathbb{R}^m$. Then there exist an $\alpha>0$ such that $$||T(x)||\geq \alpha||x||$$ for all $x$. Let $S$ a linear transformation ...
2
votes
3answers
199 views

Prove that there are no such positive integers $a,b,c,d$ such that $a^2 + b^2 = 3(c^2 + d^2)$

Prove that there are no positive integers a, b, c, d such that $a^2 + b^2 = 3(c^2 + d^2)$. Hint: What can you say about divisibility of a and b by 3? Look at solution with smallest possible a.
1
vote
1answer
89 views

Covering chessboard with L-tetrominoes

Consider an n x n chessboard with all 4 corner squares removed. Prove that if the board can be covered with L-tetrominoes then n-2 is a multiple of 4. Is the converse true? (an L-tetromino is a plane ...
0
votes
1answer
40 views

Proving that a log is at most $(n^p)$

I am trying to show for all $b > 1$, and all $p > 0$, $log_b(n) = O(n^p)$. I am trying to use the theorems that for all $p$ greater than or equal to $1$: i)$\quad log_2(n)=O(n^{\frac{1}{p}})$ ...
6
votes
2answers
111 views

Prove $(3x^2+3) \geq (x+1)^2+1$

$(3x^2+3) \geq (x+1)^2+1$ I tried using a direct proof but I think I got stumped along the way. $3x^2+3 \geq x^2+2x+2$ $2x^2+1 \geq 2x$ $2(x^2) +1 \geq 2x$ $x^2 + (1/2) \geq x$ How can I make ...
0
votes
1answer
65 views

Convergence of formal power series substitution

Prove that the substitution of formal power series $F(G(x))=\sum_{k\geq0}f_k \frac{G(x)^k}{n!}$ converges for every $F$ if and only if $G(0)=0$
0
votes
1answer
110 views

Prove that a Language is Non-Regular Using Closure Properties

Use the closure properties of regular languages and a language $B$ known to be non-regular to prove that a language $A$ is not regular. My understanding is that the closure properties only apply when ...
1
vote
1answer
108 views

Question about the infinite products of formal power series

I need a proof for this: Let $(F_j)_{j\ge 0}$ be a sequence of formal power series. The infinite product $\prod_{j\geq0}(1+F_j(x))$, where $F_j(0)=0$, converges if and only if ...
2
votes
2answers
82 views

Question that can not be solve analytically .

You can know that the solution of this non-linear simultaneous equations is y=2 and x=3; but the question is : How can mathematically ( algebraically ) find this. \begin{array}{lcl} x^y & = & ...
0
votes
1answer
99 views

Natural deduction proof of a simple formula.

I am trying to prove the following formula using only natural deduction system: $$\vdash (A \supset (A \supset B)) \supset (A \supset B),$$ and this, according to natural deduction rules leads to $$A ...
0
votes
1answer
39 views

Trigonometric Prοof

How can one show that for any angle $\theta$ such that $0<\theta<2\pi/5$ the following equation is true? $\sqrt{1-\cos\theta} ...
3
votes
2answers
452 views

Formal power series, the Chain Rule and the Product Rule.

Definitons Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n : a_n \in \mathbb{C} \right\}$$ be the set of formal power series of $x$. Exercise i) If $F_1(x)$ and $F_2(x)$ are power series ...
1
vote
1answer
53 views

A plane is a surface.

I am trying to show that a plane is a surface. I posted what I did. I find $σ$ But I cannot find $σ^{-1}$. Also as I said, I need to verify $σ$ is 1-1 continuous and continuous inverse. How can I do ...
1
vote
1answer
55 views

Let $p>q$ be primes such that $p \equiv 1 \pmod q$. Then there exists exactly one (up to isomorphism) abelian group of order $pq.$

Let $p>q$ be primes such that $p \equiv 1 \pmod q$. Could anyone advise me on how to show there exists exactly one (up to isomorphism) abelian group of order $pq$? Thank you.
1
vote
2answers
91 views

Topology - Composition of two isometric embeddings

Prove that the composition of two isometric embeddings is an isometric embedding and that the composition of two isometries is an isometry. I have been working on this problem for sometime, if ...
6
votes
1answer
1k views

Proof Involving a Problem from “Good Will Hunting”

I don't know if any of you have seen the movie "Good Will Hunting" but there is a particular mathematics problem in the movie that is of interest to be. One of the problems used in the movie is "Draw ...
2
votes
1answer
153 views

Application of Sylow's theorem

Let $p>q$ be primes. $ (1): \exists $ non-abelian group of order $pq$ $\Longleftrightarrow$ $p \equiv 1 (mod \ q)$ $(2):$ Any $2$ non-abelian groups of order $pq$ are isomorphic to each other. ...
2
votes
1answer
420 views

Showing that a superset $B$ of a set $C$ is connected if $B \subset \bar C$.

I would like feedback on the following proof I just wrote up: Let $C$ be a connected subset of the metric space $X, d$ and let $C \subset B \subset \bar C$. Prove that $B$ is connected. Let $B, ...
2
votes
1answer
45 views

Functions, Continuity and IVT

Suppose that $g$ is a function defined and continuous on $\mathbb{R}$ and $n$ is a positive integer such that $$\lim_{x\to \infty} \dfrac{g(x)}{x^n} = 0 = \lim_{x\to -\infty} \dfrac{g(x)}{x^n}$$ (i) ...
0
votes
2answers
46 views

Functions and the IVT

Let $g, h$ be continuous functions defined on some interval $J$ and suppose that $g(x) \neq 0$ for any $x \in J$. If $g(x)^2 = h(x)^2$ for all $x \in J$, show that either $g(x) = h(x)$ for all $x \in ...
-1
votes
2answers
91 views

Deterministic Finite Automata with finite strings

How could I prove that every language with a finite number of strings is the language of some DFA?
1
vote
2answers
38 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
3
votes
1answer
133 views

If $x\le c\le y$ and $|x-y|<\delta$, then $|\frac{g(x)-g(y)}{x-y}-g'(c)|<\epsilon$

Alright, so here goes the question: Consider a function $g$ that is differentiable at some point $c$ on an interval $[a,b]$. Prove that: $\forall$$\epsilon$ $>0$ $\exists$$\delta$ $> ...
1
vote
2answers
58 views

Proving a trigonometric identity.

I really need some help with this question. I need to prove this identity: $$\frac{2\sin^{3}x}{1-\cos x} = 2\sin x + \sin 2x.$$
1
vote
0answers
82 views

Show that Weyl algebra is noetherian

Let $k$ be a field. I want to show that the ring $D=k\left[x_1,x_2,\dots,x_n,\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\dots,\frac{\partial}{\partial x_n}\right]$ which acts on ...
0
votes
1answer
69 views

How do I prove a convolution is a polynomial?

I have a problem with the following question: $f$ is a continuous function and $f=0$ when $x\notin [0,1]$. $(g_n)=x^n$ when $-1<x<1$ and $0$ else. Show that $(f*g_n)$ is a ...