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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0answers
13 views

Find the $\alpha_i$ is this decomposition of $S_4$

We have seen in class he Cayley graph of the cube group (that we saw that it is isomorphic to $S_4$) with respect the generators $g_1$=A half of a turn in the diagonal and $g_2$= one third of a turn ...
3
votes
4answers
76 views

Show that $S$ and $2^{S}$ are not equinumerous. (Not bijective?)

I have tried to look for a problem the same as mine, but I have not been too lucky, or if I did I had trouble applying that solution to my problem. Any help would be appreciated. I know how to solve ...
0
votes
1answer
37 views

Proof of a Cayley graph result.

I am searching now for the most explicative and easy understand in this level (I am a second year student) proof of the following result: A Cayley graph is connected iff S generates G where G is a ...
2
votes
4answers
70 views

If $a$ is an odd integer then $x^2+x-a = 0$ has no integer solutions

I'm suppose to prove by contrapositive that if $a$ is an odd integer then the equation $x^2+x-a=0$ has no integer solution. By contrapositive: If the equation $x^2+x - a = 0$ has an integer ...
1
vote
1answer
49 views

Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$.

Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$. Here is what I have tried: Let $a \le b$. Multiplying both sides of this inequality by $a$ results in $a^2 \le ab$. It follows that $a ...
4
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3answers
184 views

Formal logic behind defining variables in a proof (not E.I. or U.G, etc.)

This is something I have been curious about and hopefully has a simple answer. Often when looking at proofs I will come upon a step that goes along the lines of "define y = ..." and then proceed to ...
2
votes
0answers
46 views

Numbers in the form $10^n + 1$ with square divisors

Basically, describe every number in the form $10^n + 1$ with square divisors meaning at least one of it's divisors is a square. Of course, there's infinite, but give a general algorithm for finding ...
0
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0answers
32 views

Proof sketch for a convex function, help.

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
3
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1answer
45 views

How to show that if there's a fast matrix inversion algorithm, then there's a fast multiplication algorithm?

Is there a way to show this and vice versa? Suppose $F_n$ is the number of flops required by some algorithm to perform the inversion of an $n-by-n$ matrix. Assume that there exists a constant $c_1$ ...
1
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3answers
50 views

Find the Range and Domain of the following function

The function is: $f(x,y) = \frac{2}{\sqrt{3-x}} + \frac{1}{\sqrt{4-y}}$ I have found the domain and the Range intuitively. But how would I formally prove that my assumption of the Range and Domain ...
4
votes
6answers
163 views

Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$ [duplicate]

Having the following inequality $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$$ To prove it for all natural numbers is it enough to show that: ...
0
votes
3answers
231 views

Let A and B be sets. Prove that A = B iff the power set of A is equal to the power set of B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! Let A and B be sets. Prove that A = B iff the power set of A is equal to the power set of B. Assume that A and B are ...
0
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3answers
37 views

For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B. ...
1
vote
1answer
130 views

Prove that if x ∉ B and A ⊆ B, then x ∉ A

I am an undergraduate student and I am wondering if the strategy and the writing of this proof are correct. Please help me! Prove that if x ∉ B and A ⊆ B, then x ∉ A. Assume that x ∉ B and A ⊆ B, ...
2
votes
1answer
96 views

Can the chain rule be proven by math induction?

I need to prove the chain rule for a math project and I am wondering if it can be proven by math induction. If not, how can this rule be proven?
0
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0answers
18 views

How can I prove the relationship between the integrals of $f(x)$ and $f(x-c)$?

I need to prove $\int_{a+c}^{b+c}f(x-c)dx = \int_a^bf(x)dx$. How would I start this? Would breaking up the integral into $\int_{a+c}^bf(x-c)dx + \int_b^{b+c}f(x-c)dx$ help at all?
1
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0answers
55 views

Computation of the probability of Bernoulli trials

I want to solve the following : Independent trials that result in a success with probability $p$ and a failure with probability $1-p$ are called Bernoulli trials. Let $P_n$ denote the probability ...
2
votes
0answers
42 views

Proof of an inequality about primes

I'm very new to number theory and looking for a proof of the following inequality: $$c' \log^{\text{#} \mathbb{P}}{R} \leq \sum \limits_{\substack{n \leq R\\p|n \implies p \in \Bbb P}} 1 \leq c ...
2
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3answers
72 views

Suppose $f:\mathbb{R} \to \mathbb{R}$ is differentiable and $f'(x) \geq c > 0, \forall x.$ Then $f(\mathbb{R}) =\mathbb{R} \ $

Suppose $f:\mathbb{R} \to \mathbb{R}$ is differentiable and there exists $c>0$ such that $f'(x) \geq c, \forall x.$ Could anyone advise me how to prove $f(\mathbb{R}) =\mathbb{R} \ $ I have ...
0
votes
2answers
54 views

Squaring any element of the empty set.

I am asked to prove that when squaring any element of the empty set, one should always get zero. Of course the empty set is the set which contains no elements. If you square nothing then you should ...
1
vote
1answer
51 views

Generalising mean value theorem for integrals

I want to generalize mean value theorem for Riemann integrals to $\Bbb R^n$, but I do not know how to formulate it. Can someone please help me with formulating the theorem? I think I can prove it ...
0
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1answer
79 views

Matrix Invertibility: AB invertible implies A,B invertible

AB invertible $\implies$ AB is the product of elementary matrices $\implies$ A, B are the product of elementary matrices $\implies$ A,B are invertible, since the products of elementary matrices are ...
2
votes
3answers
180 views

Matrix Invertibility- Verify Proof

$$ \text{ if $A,B$ are invertible matrices, prove $AB$ is invertible } $$ $$ A, B \text{ invertible} \implies \text{ $A$ and $B$ are each the product of elementary matrices } E_{1} E_{2} \cdots ...
0
votes
3answers
41 views

How to prove the relation between the integral of $f(x)$ and $f(-x)$?

How can I prove $\int_{-a}^0f(x)dx=\int_0^{a}f(-x)dx$? I can't say anything about the relationship between $f(x)$ and $f(-x)$ without knowing whether it's an even or odd function. The problem says to ...
1
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1answer
37 views

$ P$ $ \Rightarrow$ ($Q$ or $R$) is not equivalent to $P, \neg Q\Rightarrow R$?

Suppose we wish to prove this implication $ P$ $ \Rightarrow$ ($Q$ or $R$): So,we suppose that $P$ is true and then we try to show that $\neg Q \Rightarrow R$ ;to do that we suppose that $ \neg Q $ ...
3
votes
1answer
57 views

Show that $\,f(x) = \sum_{n=0}^\infty a_nx^n$, for $x \in [0,1]$, is of bounded variation

Let $\{a_n\} \subset \mathbb{R}$, be such that $\sum_{n=0}^\infty \lvert a_n\rvert < \infty$. Define $$f(x) = \sum_{n=0}^\infty a_nx^n \quad \text{for } x \in [0,1]$$ Prove that $f$ is of bounded ...
0
votes
2answers
103 views

Probability of failing a test or passing, different scenarios.

The probability to fail a drive test is $t$ while the probability to pass a drive test is $t+0.2$. I really need to know if what I am doing is correct. I guess I should be using bernoulli but I don't ...
23
votes
1answer
666 views

Moriarty's calculator: some bizarre and deceptive graphical anomalies

Background: This is a problem I first came across a few years ago in a calculus textbook (a James Stewart one), where it addressed some of the pitfalls of using graphing calculators. The original ...
6
votes
1answer
37 views

Show that if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$

Prove if $x^2 y=2x+y$, then if $y \neq 0$ then $x \neq 0$. Obviously, $x,y \in \mathbb R$. I know this is rather simple. It is more about the process than this example. Is it logically correct to do ...
0
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3answers
150 views

Prove that if $(x+y)$ is even, then $(x-y)$ is even, for integers.

I am completely stuck on how to prove this. I need to prove it multiple ways too (directly, contrapositive and by contradiction). My problem is that I can't figure out how to isolate $x$ and $y$ in ...
1
vote
5answers
45 views

Two Questions: (1) Is My Proof Method Legitimate, and (2) How Might Proof by Cases be Accomplished?

Prove 5x-4 is even iff 3x+1 is odd. I have two questions: First, for example, could we assume that 5x-4 is even such that 5x-4 = 2k, for some integer k and then manipulate the above equation to ...
0
votes
3answers
147 views

Proof-making for “A Set is a Subset of Itself” / Law of Identity.

Recently I've been trying to figure out a proof regarding set theory, for the following theorem: "A set is a subset of itself" or $∀x:S ⊆ S$, or: $∀x: (x∈S ⟹ x∈S)$ ProofWiki states that such a ...
1
vote
1answer
35 views

how to prove that this function is well defined?

I am working with intervals on the real line. The interval can be of any type $[a,b],(a,b),[a,b)$ etc. For every interval, I define the length $l(I)=b-a$. Let A be a set of finite disjoint ...
1
vote
3answers
65 views

If p is an odd prime, prove that $a^{2p-1} \equiv a \pmod{ 2p}$

Let $m = 2p$ If p is an odd prime, prove that $a^{2p - 1} \equiv a \pmod {2p} \iff a^{m - 1} \equiv a \pmod m$. I have no idea on how to start. I was trying to find a form such that $a^{m - 2} ...
8
votes
6answers
147 views

Prove that $(n+1)^{n-1}<n^n$

How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$ I have tried several methods such as induction.
-1
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2answers
87 views

Need help with starting a proof of linear independence [closed]

Let $\mathbb{R}^m$, $\mathbb{R}^n$ be Euclidean spaces. Let $T: \mathbb{R}^m → \mathbb{R}^n$ be a linear transformation with corresponding $n × m$ matrix A. Let $x_1, x_2, ..., x_k \in \mathbb{R}^m$ ...
0
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2answers
57 views

Proof : Do 4 days fall on the same day?

I was working my way through some discrete math proof examples from Discrete Math by Rosen and being a newbie am stuck on this problem : Show that at least four of any 22 days must fall on the ...
2
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2answers
162 views

Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x+y=z

This is a homework assignment, please tell me if my proof is correct! Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x + y = ...
0
votes
2answers
49 views

if $2a+3b \geq 12m+1$, then either $a \geq 3m+1$ or $b \geq 2m+1$

Not sure how to go about proving this. So far I've declared the contrapositive but can't seem to get further... Let $a,\ b$ and $m$ be integers. Prove that if $2a+3b \geq 12m+1$, then $a \geq 3m+1$ ...
0
votes
0answers
15 views

Prove that the solution of the differential equation exists for $0\le x\le min(a,{b\over a^2+b^2})$

Consider the initial value problem $y'=x^2+y^2, y(0)=0$ and let $R$ the rectangle $0\le x\le a$, $-b\le y \le b$. Prove that: The solution $y(x)$ exists for $0\le x\le min(a,{b\over a^2+b^2})$ I ...
3
votes
1answer
109 views

The number of regions into which a plane is divided by n lines in generic position [duplicate]

Suppose that $n$ lines are drawn on a plane in such a way that no lines are parallel and no three of them intersect at a point. Let $r(n)$ be the number of regions the plane is divided into after ...
0
votes
2answers
93 views

Prove that det(AB) = det(A) det(B) in AB ∈ $GL_2(\mathbb{R} \!\,)$

Prove that det(AB) = det(A) det(B) in AB ∈ $GL_2(\mathbb{R} \!\,)$. Use this result to show that the binary operation in the group AB ∈ $GL_2(\mathbb{R} \!\,)$ is closed; that is, if A and B are in ...
2
votes
1answer
104 views

Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction

My homework has this problem: Prove that there exist irrational numbers $x$ and $y$ such that $x + y$ is rational. There is an easy solution that I found on mathbitsnotebook.com: ...
4
votes
1answer
52 views

Uniform Convergence Analysis

Uniform Convergence of $n^{2}(x)^{3}e^{-nx^{2}}$ on $[0,1]$ My attempt: criterion: suppose $f_n:I\to\ J$ is a sequence of functions which converges point wise to a function $f$, then the ...
2
votes
0answers
22 views

Is this (sketch of a) proof sound? Uniform cont.

I want to show that a specific trigonometric function is not uniform continuous, as far away from 0, it oscillates like crazy. What I (think I) want to show: I can find an $\epsilon$ such that ...
0
votes
2answers
56 views

Linear Transformation between Isomorphic Vector Spaces

Suppose $f:V\to W$ is a linear transformation, and that there exists a basis $\{v_1,\ldots,v_n\}$ for $V$ such that $\{f(v_1),\ldots,f(v_n)\}$ is a basis for $W$. Prove that $f$ is an ...
0
votes
1answer
19 views

Let $x_{0}=1$ and $x_{1}=-1$ For $n\geq0$ inductively define $x_{n+2}=x_{n+1}+6x_{n}$

I am not so sure how to do this problem and would like some help here. How would you induct a relation given this information here? I mean I know what induction means but I'm not so sure what I'm ...
0
votes
0answers
30 views

Proofs with GCD are equal with a and b

I need to prove that if $a$ and $b$ are negative then $gcd(a,b) = 2gcd(\frac{a}{2},\frac{b}{2})$ I feel like it should be more complicated than $2\cdot \frac{a}{2} = a$, so it's the same GCD.
0
votes
1answer
58 views

Using contrapositive to Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [duplicate]

so I know that the contrapositive will be something like; If all the numbers are greater than or equal to 7, then the average cannot be less than 7. How do i go about proving it from there? or is ...
2
votes
2answers
80 views

Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [closed]

I tried proving this by contraposition, by saying, "If every number that is being averaged is greater than 7, then the average of a thousand numbers is less than 7." This seems easier to prove, but I ...