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.

learn more… | top users | synonyms

3
votes
5answers
120 views

Simple question with a paradox

"I have three boxes, each with two compartments. One has two gold bars One has two silver bars One has one gold bar and one silver bar" You choose a box at random, then ...
0
votes
1answer
55 views

What is the wrong in proving this Assumption?

In the famous case of proving that total number of degrees in a graph $G$: $\sum \deg(v_G) =2m$. By Using Proof by induction:- for: $$m=0: 2m= 2*0 =0 \tag 1$$ is true .. $(2)$...We add a new edge to ...
0
votes
3answers
77 views

Show that: $\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$

I am fully uncertain of how to approach this problem: Show that: $$\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$$ We've just completed the section on improper ...
1
vote
1answer
53 views

Methods to prove that a function is continuous

Although I seem to understand the concept of continuity in connection with functions, I am often stuck proving that particular functions are continuous. I think the epsilon-delta definition is the ...
8
votes
3answers
333 views

Need Suggestions for beginner who is in transition period from computational calculus to rigorous proofy Analysis

I have completed basic calculus 1,2,3 courses, Linear Algebra, etc. I have not, however, got into rigorous Analysis yet, which I am planning to do now. I have three books in mind. They are : Terence ...
0
votes
0answers
17 views

Functions with 0 as an interior point of the domain

We have $f,g,h : \mathbb{R} \rightarrow \mathbb{R}$, with $0$ an interior point of the domain. Let $k \in \mathbb{N}$. We use the following notation $h = \mathbb{O}(x^k)$ if the open interval $ I ...
1
vote
1answer
23 views

Intermediate theorem on a function of more dimensions

I have the following question in my textbook: Let $M>0$ and $\phi : \mathbb{R}^p \rightarrow \mathbb{R}$ be differentiable with the property that $$||\nabla \phi(x)|| \leq M$$ Proof that ...
0
votes
1answer
38 views

Riemann integral of a non continuous function

We have a function $f : I=[0,1] \rightarrow \mathbb{R}$ defined as: $$f(x)=\begin{cases} 1 &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ a) Show ...
0
votes
1answer
63 views

Riemann Integral

I tried to do the following excercise Let $a,b \in \mathbb{R}, a<b$. We have a bounded function $ f: [a,b] \rightarrow \mathbb{R}$ which has an integral or in other words, there exists a ...
0
votes
1answer
34 views

If $G$ is simple and $deg_+(v) \ge k\ge 1$ , then there is a simple cycle of at least size $k+1$

I am going to show you my proof/ and please correct me if wrong: Begin with some node $v$, and mark it. Follow one of its outgoing edge $(v,w)$ to next unmarked node, and mark it, by doing this ...
0
votes
1answer
39 views

Strong Induction: Prove that sqrt(2) is irrational

This question comes directly out of Rosen's Discrete Mathematics and It's Applications pertaining to Strong Induction. Use strong induction to prove that $\sqrt{2}$ is irrational. [Hint: Let $P(n)$ ...
3
votes
1answer
40 views

Arrange 1-12 around a circle

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with plain old algebra, which yields the shortest, simplest proofs, but other ...
2
votes
2answers
41 views

Proofs utilizing the Well-Ordering Property

This question comes directly from as an example in Chapter 5.2 of Rosen's Discrete Mathematics and It's Applications textbook on page 341. Use the well-ordering property to prove the division ...
-2
votes
2answers
36 views

Proof related to matrix [duplicate]

Let $A$ and $B$ be $n \times n$ real matrices such that $A^2 = I, B^2 = I$ and $(AB)^2 = I$. Prove that $AB = BA$. Someone help me with this problem
0
votes
1answer
32 views

Proof related to matrix with if and only if condition

Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined. Prove that $$(A+B)^2=A^2+B^2+2AB\quad\text{if and only if}\quad AB=BA.$$ Someone help me with this.
2
votes
5answers
75 views

Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$ [duplicate]

So here the problem goes: Show that if $x \ge 1$, then $x+\frac{1}{x}\ge 2$. This is a very interesting word problem that I came across in an old textbook of mine. So I know it's got something ...
0
votes
0answers
16 views

Proofs By Induction Help

Hey I'm having some problems on these proofs. I think Im doing right but if anyone can show me the right way to do them that would be great! 1) ∑ i=1, n of (2i) = n^2 + n ...
11
votes
1answer
205 views

How to combine the four Theorems in order to prove the statement?

I have a question concerning a statement about Random Walks on $\mathbb{Z}$. Let $F$ be a distribution on $\mathbb{Z}$ which has mean $0$ and finite variance. Let $\left\{X_1,X_2,\ldots\right\}$ be an ...
2
votes
2answers
34 views

Proving binary integers

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with binary integers (For ${0, 1, 2, 3}$ we have the representations $0, 1, 10, ...
14
votes
8answers
1k views

Variation on Pythagoras: If $a^2 + b^2 = c^2$, then $a + b \leq c\sqrt{2}$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of the Pythagorean Theorem using calculus, trigonometry, ...
0
votes
1answer
82 views

Proving that $\sum \deg(v) = 2m$ for any Graph $G$

Here is My proof, please correct me if wrong, I try to be formal. Proof by Induction: Let $\sum \deg(v)=2m$ assumption... when #of nodes is $n=0$. so here the equation is $\sum \deg(v)=2(0)=0$ ...
0
votes
2answers
74 views

Explain how the proof is done

A solution of matrix problem appears to be as follows some one explain the following in the solution why is A cube is eliminated and fourth power of A is obtained? In the seventh line In the ...
1
vote
6answers
122 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
6
votes
2answers
115 views

What does “rigorous proof” mean?

I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more ...
0
votes
1answer
40 views

Help with a (simple?) proof

I have the following problem: I want to prove under what conditions the value of a cell (e.g. 1 or a in the graphics) is the column mean (e.g. 1.5) plus row mean (2) minus the grand mean (2.5) like ...
0
votes
1answer
42 views

Prove that the hypotenuse is the longest side in a right triangle. How to write a formal proof for something so obvious?

From trig text. Given hint: is $ a^2 + b^2 > a^2 $. Pythagorean theorem,obviously. What would be an acceptable proof?
1
vote
1answer
35 views

Maximal Distance of a graph

So I have the following graph $G = (V, E)$ with $V = [d]^n$ and $E = \{\{(a_1,\dots , a_n), (b_1, \dots , b_n)\} | a_2 = b_1 a_3 = b_2, \dots a_n = b_{n-1}\}$. That is called a $(n, d)$-dimensional ...
1
vote
5answers
91 views

Prove that $ a^2-4b \neq2$ if $ a,b \in \mathbb{ Z}$

My solution : We suppose that is true. Then by contradiction: $a^2-4b-2=0$ $a^2=4b+2$ $a=2(b+1/2) ^{0.5}$ then $(b+1/2)$ is fraction and rooted by $0.5$ so the square root of any fraction $+$ ...
-4
votes
1answer
113 views

I have proved that 1 + 1 = 0 [closed]

I have proved that 1 + 1 = 0 in one of my questions where a field was given. I was wondering if it is true in every field we have 1 + 1 = 0. Also i was wondering (i know how to prove 1 + 1 = 0) can ...
-1
votes
2answers
47 views

Every field has at least two elements

I got a question saying in every field (F, +, ⋅, 0, 1), the set F has at least 2 elements. It asks if it is true prove it or if false provide a counterexample. I understand the idea of finite fields ...
0
votes
2answers
44 views

Field Question Proof with Axiom 4

Prove that if $(F,+,⋅,0,1)$ is a field, then there is no element $w ∈ F$ such that $0 \cdot w = 1$. Note that Axiom 4 from lecture (aka "M4" in the textbook) ensures that for $x ≠ 0$, there is a $w ∈ ...
1
vote
1answer
53 views

Proof of definition of invertible matrices [closed]

Let $A \in \Bbb R^{n \times n}$. Then $A$ is invertable if and only if a Matrix $B \in \Bbb R^{n \times n}$ exist such that $AB=E$. This seems like the definition of an invertible matrix but how ...
0
votes
0answers
30 views

Determining if two bounds are true

Question says assume $f$ and $g$ have a domain of the integers, and target space of the real numbers. $f$ and $g$ are bounded. Prove if the following statements are true or give a counterexample: if ...
2
votes
1answer
132 views

Strategy/Proof behind the Perfect solution of a Multiplication Game

So the below is the question Question: Jacob and Vicky play the fun game of multiplication by multiplying an integer p by one of the numbers 2 to 9. Jacob always starts with p = 1, does his ...
2
votes
2answers
45 views

Proof by contradiction and mathematical induction

$\sum_{i=1}^n {2\over3^i}={2\over3}+{2\over9}+\dots+{2\over3^n}=1-{({1\over3})^n}$ I had this problem in class and we proved using 2 different methods: contradiction and mathematical induction. I ...
1
vote
1answer
39 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
3
votes
2answers
52 views

Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.

Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $. For this question, I think that I would use the completeness axiom. As $3$ is greater than $2$, so $S$ has a ...
-1
votes
1answer
78 views

f and g are bounded with domain of integers and target the real numbers . If f/g is bounded, then g/f is bounded.

I have come up with two bounded functions f = 1/x^2+1 and g = 1/x^2+2 and these tell me that g/f is also bounded. However, I am having trouble writing a proof or proving that g/f is not bounded by ...
-1
votes
1answer
80 views

f and g are bounded . if 1/g is bounded, then f/g is bounded.

I would like some help understanding how to go about this question. I think that f/g is not bounded, but I cannot figure how to show that f/g is not bounded.
0
votes
1answer
81 views

Proving formulas with products of Fibonacci numbers

While digging through my old notes, I stumbled upon some formulas involving multiplication of Fibonacci numbers that I discovered about 7 years ago (being fascinated with Fibonacci numbers at the ...
2
votes
0answers
55 views

Help in a proof in Fulton's algebraic curves book

I'm reading Fulton's algebraic curves book and I didn't understand this proof of proposition 7 (page 106) very well: So I have the following doubts: I didn't understand why $\text{ord}_P(f')\ge ...
0
votes
1answer
27 views

How to prove a recursive sequence is bounded

I have $f(x+1)-f(x)=\dfrac {-1}{x^2+x}$ and I want to prove it is bounded: $f(x)< 1$ for all $x>0$. If f(1)=1 I've tried induction but no luck.
2
votes
0answers
38 views

Find Solution regarding 2-Norm

I try to understand that, but I have no clue what do to and how to do it. $A$ is a $m \times n$ matrix with $rg(A)=m$. Find the solution for $Ax = b$, which is regarding to the $2$-norm (I guess ...
0
votes
1answer
15 views

Understanding part of a proof, to show commutatitivy of geometric mean for matrices

Ando defined a matrix geometric mean, for two positive $n \times n$ matrices, as follows: $$G(A,B) = B^{1/2}(B^{-1/2}AB^{-1/2})^{1/2}B^{1/2}$$ where all square roots are positive square roots. This ...
3
votes
5answers
73 views

Prove that $(1+a)^x>1+ax$ when $x>1$ and $0<a<1$

Prove that $(1+a)^x>1+ax$ when $x>1$ and $0<a<1$ and $(1+a)^x<1+ax$ when $0<x<1$ and $0<a<1$ I was trying to do it the usual way which is to consider the function ...
1
vote
1answer
23 views

Can we construct a successor function from successive applications of those two functions?

Let $f(x) = 5 \cdot x + 3$ and $g(x) = \frac{x}{8}$, Is it possible to construct a function $s$ such that $s(x) = x + 1$ via successive applications of any of f and g?
0
votes
0answers
12 views

Property of the local solution of a static constrained optimization problem

In Nocedal/Wright's Numerical Optimization (1999, 1E) on p. 332 in subsection Feasible Sequences of section 12.3 Derivation of the First-Order Conditions they claim that a local solution $x^*$ to a ...
1
vote
1answer
52 views

Clifford Theorem as an easy corollary of Riemann-Roch Theorem

I'm studying Fulton's algebraic curves book and on page 109 he proves the Clifford's theorem: I have these doubts: 1.Why does he consider only the divisors $D\ge 0$ and $W-D\ge 0$? 2.What ...
0
votes
0answers
29 views

How can I show that $x'(t) = - (1-x(t))'$ from $x(t)$'s functional form?

I have a differential equation modelling the concentration of vaccinators in a population. Here are a few assumptions. Assume we can write the concentration of vaccinators as $x$, and that we can ...
0
votes
1answer
28 views

Is there some set of 2-d points so that there does not exist a function that connects them?

Let's say I have a set of points $\{(x_0,y_0),\ldots,(x_n,y_n)\}$. Does there exist some function $f$ so that $f(x_i) = y_i$? If so, how can I prove that for any arbitrary set of points there exists ...