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
73 views

Is This a Proof by Induction?

Prove, disprove, or give a counterexample: $$\sum_{i=0}^n \left(\frac 3 2 \right)^i = 2\left(\frac 3 2 \right)^{n+1} -2.$$ I went about this as a proof by induction. I did the base case and ...
3
votes
2answers
70 views

Is a quadrilateral with one pair of opposite angles congruent and the other pair noncongruent necessarily a kite?

If convex quadrilateral ABCD has congruent angles A and C and the other pair of angles B and D are not congruent, is ABCD necessarily a kite (two pairs of consecutive congruent sides but opposite ...
0
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2answers
26 views

Prove that $A \cup C \sim A$ when $C$ is countable and $A$ is infinite.

First of all, if $A$ is countable, the result is true because the union of countable sets is countable. If $A$ is uncountable, and I have no idea how to prove this part. Can it be useful? is there a ...
4
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1answer
45 views

Problems with a proof that -in a linear order- a minimal element is the smallest element

I have a problem with a proof I found in Velleman's "How to prove it". This is sort of interesting, because it is the very first time I cannot see the structure of a proof presented in the book. The ...
1
vote
2answers
33 views

Associates in Domains

Let D be a domain and $a, b \in D^*$. Show that $a$ is a proper divisor of $b$ if and only if $b=ax$ for some nonzero nonunit $x$. I'm just really not sure how to start this. Any advice would be ...
0
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0answers
17 views

Ring Theory Domain Proof

Let D be a domain. Show that $D[X]^x$=$D^x$. Because D is a domain it means that it is cancellative and D has no nonzero zero divisors. The only units in $D[X]^x$ are the units in $D^x$ so it's ...
1
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0answers
62 views

Noetherian Ring and Homomorphic Image

Prove that, if $R$ is Noetherian, then so is each homomorphic image of $R$. I know that by the Fundamental Homomorphism Theorem this is the same as showing that if $R$ is Noetherian, then so is ...
3
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3answers
105 views

Let $c$ and $d$ be real numbers, not both zero, and let $f(x) = (ax + b)/(cx + d)$

Let $c$ and $d$ be real numbers, not both zero, and let $f(x) = (ax + b)/(cx + d)$. Then $f$ is a function $S\to\mathbb R$ where $$ S = \{ x \in\mathbb R | cx + d \neq 0\}. $$ Under what conditions ...
3
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2answers
101 views

Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.

This is a recent homework bonus question assigned in my Proofs and Conjectures class. It (evidently) includes and evaluates our understanding of elementary-set theory and how to determine and prove ...
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3answers
91 views

Proofs Homework Help

I have been struggling with my proofs homework this week and would greatly appreciate any help. Prove, disprove, or give a counterexample: Suppose $f:X\to Y$, $A\subseteq Y$, $B\subseteq Y$ and ...
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0answers
28 views

Determine weaker hypotheses for Evaluation Fundamental Theorem of Calculus (Abbott p 202 T7.5.3)

(p 200 T7.5.1) If $f:[a,b] \to R$ is integrable and $F:[a,b] \to R$ satisfies $f(x)= f'(x) \; \forall x \in [a,b]$, then If $g$ is integrable on $[a,b]$, then $\int_a^b f = F(b) - F(a)$. ...
2
votes
1answer
153 views

Proving that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$ for all positive $n$

Hello I am having difficulty proving the following inequality: $$ \sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction ...
25
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12answers
4k views

Prove that a counterexample exists without knowing one

I strive to find a statement $S(n)$ with $n \in N$ that can be proven to be not generally true despite the fact that noone knows a counterexample, i.e. it holds true for all $n$ ever tested so far. ...
2
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2answers
81 views

Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)

1. How can we presage to use Mean Value Theorem to start the proof? 2. Mean Value Theorem engenders a point in an open interval. Shouldn't this be $x_i \in (t_{i - 1}, t_i) $? After ...
0
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3answers
93 views

Derive Closed form sum of N^2

Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N ex ...
0
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1answer
25 views

Floor and Ceiling question

This was a homework question. I wasn't able to get far because I couldn't determine the properties of floor and ceiling functions. Any help would be awesome. $\def\lc{\left\lceil} ...
2
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0answers
58 views

Proof of two properties of a simple math function

I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function: ...
0
votes
1answer
34 views

Prove something that is differentiable

The question states If g(x) is differentiable, then for any positive integer $n$, $(g(x))^n$ is differentiable and $\frac d{dx}$$(g(x))^n=(g(x))^{n-1}g'(x). $ Where does the continuity of g enter ...
0
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2answers
58 views

Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$

Hello I have been blasting at this inequality proof and it is just not doing what I want it to do: Prove that $2^{n-1}(a_1a_2\ldots a_n + 1) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ assuming that ...
2
votes
1answer
35 views

Inductive proof of an inequality

I am trying to prove this inequality by induction: For all $x$ in the interval $x\in [0, \pi]$, prove that: $$ |\sin (nx)| \leq n\sin(x) \textit{, n a nonnegative integer}$$ The base case is ...
2
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0answers
84 views

Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
0
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0answers
28 views

Ring Embeds in Monoid Ring

Let $(S,+)$ be a nontrivial commutative monoid and $R$ be a ring. Prove that $R$ embeds in $R[X;S]$ via $a \to aX^0$ I'm not exactly sure how to approach this... I think I may need to use the fact ...
0
votes
0answers
28 views

Hilbert tenth problem unsolvability

Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. How ...
1
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1answer
28 views

Modified Collatz Problem

How can one prove for all $n \in \mathbb N$ that the following sequence always results in $1$: Choose $m$ $x_1 = m$ $$x_{n+1} = \begin{cases} x_n/2 & \text{if $x_n$ is even} \\ \\ x_n + 1 & ...
2
votes
2answers
73 views

Proving either $x^2$ or $x^3$ is irrational if $x$ is irrational

I had a test today in discrete mathematics and I am dubious whether or not my proof is correct. Suppose $x$ is an irrational number. Prove that either $x^2$ or $x^3$ is irrational. My Answer: ...
1
vote
1answer
51 views

Prove $f_x(0,0) , f_y(0,0)$ exist

Let $f(x,y)= (xy)^{1/3}$ Using definition of the partial derivatives, prove $f_x(0,0), f_y(0,0)$ both exist. Show that the directional derivative for $f$ in any direction other than $i$ or $j$ does ...
2
votes
2answers
53 views

How to prove $n^3 < 4^n$ using induction? [duplicate]

It's true for all Natural numbers. What I've got so far: Prove $P(0) \to $ base case: Let $n = 0$ $(0)^3 < 4^0 = 0 < 1$ Then $P(0)$ is true. Part Two: Prove $P(n) \Rightarrow P(n + 1) ...
1
vote
0answers
31 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
0
votes
1answer
78 views

The annihilator of an intersection

I know this question has been arlready asked, but as my reputation is too low I'm not allowed to post a comment, sorry for this second post. I'm asked to prove : $(W_1+W_2)^0=W_1^0\cap W_2^0$. ...
0
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3answers
35 views

A proof using Fermat's Little Theorem?

Let $p$ be prime and let $a\in Z$ such that p doesn't divide a (sorry I couldn't find the symbol for it in MathJaX). Prove that if $k$ is the smallest integer such that $a^k\equiv 1 \pmod p$, then ...
0
votes
0answers
46 views

Distance metrics with kmeans

Context: I'm trying to derive some formulas for computing the "mean" in the K-means algorithm. So given an assignment of $m$ data points to $k$ clusters, find a formula to recompute the mean of the ...
2
votes
4answers
96 views

Finding an algebraic proof for $r{n \choose r} = n{n-1 \choose r-1}$ [closed]

I can't seem figure this proof out. How are both sides equal. $$r{n \choose r} = n{n-1 \choose r-1}$$
1
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4answers
120 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
0
votes
2answers
57 views

How do I prove this bijection?

The number of $n$-digit binary numbers with exactly $k$ $1$s equals the number of $k$-subsets of $[n]$. I think i'm on the right track, but I'm confused on how to write how it's onto and 1-1. This ...
7
votes
3answers
432 views

How to find the sum of $i(i+1)\cdots(i+k)$ for fixed $k$ between $i = 1$ and $n$?

I learned that $$\sum \limits_{i=1}^n i(i+1) = \frac{n(n+1)(n+2)}{3}$$ or in general $$\sum \limits_{i = 1}^n i(i+1)(i+2) \dots (i + k) = \frac{n(n+1)\dots (n+k+1)}{k+2}$$ From a mathematical ...
2
votes
3answers
100 views

is it wrong to do this to solve an induction question

When doing an induction problem is it wrong to simply add the next variable to both sides? for example for all natural numbers $$4+9+14+19....+(5n-1)=\frac{n}{2}(3+5n)$$ assume true for k ...
2
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0answers
45 views

the sum of the first n odd integers squared proof [duplicate]

I don't know why but I can't even get my base case to work with this proof can someone help me out? Prove that $$1^2+3^2+5^2+\cdots+(2n+1)^2=\frac{(n+1)(2n+1)(2n+3)}{3}$$ when $n$ is a non negative ...
2
votes
3answers
56 views

Proof expectation of bernoulli distribution

Suppose we have: $P(X=k) = (1-p)^k p$ $$E(X) = \sum^{\infty}_{k=0} kP(X=k)= \sum^{\infty}_{k=0} kp(1-p)^k = p(1-p) \frac{1}{p^2}=\frac{1-p}{p}$$ What I do not get is the step in the equation ...
6
votes
5answers
349 views

Proof Strategy - Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero - 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 ...
0
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5answers
38 views

help solving this proof with remainders

For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$. I tried to set $n = 3+4k$ but it doesn't help. Any hints first please?
1
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5answers
117 views

How to prove that for all natural numbers, $4^n > n^3$?

This is a problem set I have, it's not a homework but it's very important practice... Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably... The ...
1
vote
3answers
69 views

Proving if $\gcd(c,m)=1$ then $\{x\in \Bbb Z \mid ax\equiv b \pmod m\} =\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$

Okay so I'm confused on how to approach this question. If $\gcd(c,m)=1$, then $S=T$ where $S=\{x\in \Bbb Z \mid ax\equiv b \pmod m\}$ and $T=\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$. I know ...
0
votes
6answers
92 views

Suppose that $x$ and $y$ satisfy $\frac{x}{2} + \frac{y}{3} = 1$. Prove that $x^2 + y^2 > 1$.

Ok , i tried to prove this via Contrapositive setting $x^2 + y^2 \le 1$. After doing some algebra i have arrived at $x \le \sqrt{-y^2}$. I'm fairly sure this isn't right. I also solved for x and y in ...
1
vote
1answer
53 views

Proof by contradiction: $(A \subset \Bbb{R}) \land (A \neq \emptyset )\wedge (A \mbox{ is bounded below } )\to \exists x (x \doteq \inf(A))$

I must proof the following: Prop.: let be $\Bbb{R}$ a complete ordered field $$\emptyset \neq A \subset \Bbb{R} \wedge A \mbox{ is bounded below } \to \exists x (x \doteq \inf(A))$$ Proof: by ...
1
vote
1answer
66 views

Suppose that 0 < a < b. Prove that $a < \sqrt{ab} < b $ and $\sqrt{ab} \leq \frac{1}{2}(a+b)$

For part 1, I have used the NOT operator on it, giving me $a \geq \sqrt{ab} \geq b$, and then tried to prove a contradiction to the assumption. I came up with $a = b$ by transitivity, which ...
4
votes
3answers
132 views

Proof: $ \lfloor \sqrt{ \lfloor x\rfloor } \rfloor = \lfloor\sqrt{x}\rfloor $.

I need some help with the following proof: $ \lfloor \sqrt{ \lfloor x\rfloor } \rfloor = \lfloor\sqrt{x}\rfloor $. I got: (1) $[ \sqrt{x} ] \le \sqrt{x} < [\sqrt{x}] + 1 $ (by definition?). (2) ...
0
votes
0answers
27 views

Shorten a proof using Galois connections

Consider a Galois connection: $f:R\rightarrow F$ is a lower adjoint of $r:F\rightarrow R$ for partially ordered sets (actually complete lattices) $F$ and $R$. We have also $f(r(g))=g$ for every $g\in ...
2
votes
1answer
35 views

Is it possible to infer this relation without calculation?

Suppose $A\sim \mathscr{E}(\alpha)$ and $B\sim\mathscr{E}(\beta)$. Is it possible to argue that: $$\beta\,\mathbb{P}(A>B)=\alpha\,\mathbb{P}(B>A)$$ without calculating $\mathbb{P}(A>B)$ or ...
1
vote
1answer
50 views

How to presage Prove by Contrapositive, for Sequential Characterizations of Limit and Continuity? (Abbott pp 106 t4.2.3, 110 t4.3.2)

Dafinguzman answered consummately this question initially but it became too long. I want to question for different beliefs. 1. $(ii) \implies (i)$ in both Theorems 4.12 and 4.19 posit sequences ...
3
votes
3answers
327 views

Prove: Number of Derangement is odd if and only if number of items is even .

let $D_n$ be a number of Derangement of n items . prove that $D_n$ is odd if and only if n is even . i was trying to use induction on the $!n=(n-1)(!(n-1)+!(n-2))$ recurrence relation but i cant ...