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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Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
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1answer
14 views

Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$.

I have been working through the following proof: Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$. Below, I have included screenshots of the ...
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14 views

Induction problem requiring to prove for any n≥2 and any sequence, we have the following claim

So i was given a question that begins like this. We will introduce a //mystery function//, $$P:N \to N$$. We don't know a formula for $$P$$ (and we won't be able to determine one!) but we do know ...
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32 views

Determine whether it is injective, surjective, bijective or neither injective nor surjective

The question i was given asked (a) Determine whether it is injective, surjective, bijective or neither injective nor surjective. (b) If you answered "bijective" in part (a) determine the ...
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Proof Mystery Function Using Induction

A mystery function, P: N $\to$ N. P satisfies the following relationship for all $a_1$, $a_2$ $\in$ N $P(a_1a_2)$ = $P(a_1)a_2$ + $a_1P(a_2)$. Armed with only this information prove that for any n ...
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1answer
15 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
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3answers
63 views

Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.

Here is the proof: Let $a,b \in \mathbb{Z}$. Prove that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. When working these problems, I do try to set them up logically. My ...
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3answers
59 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
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3answers
64 views

Proving $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$ using strong induction

The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$. Here is my proof by strong induction: Base case: $10\cdot0=0$. Let $k\geq 0$, and suppose that for any $m\leq k$ we have ...
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24 views

Estimate for the integral using convexity bound

I'm reading the proof of Hardy and Littlewood theorem in the book Analytic Number Theory, written by Henryk Iwaniec and Emmanuel Kowalski (p. 547): Theorem (Hardy and Littlewood): Let $N_0(T)$ be ...
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1answer
46 views

Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.

I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through ...
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1answer
66 views

Interesting property of Pascal's Triangle

I was looking at the Pascal's Triangle and saw that for all central numbers in even length row $a \gt 17$, the number $\dbinom{a}{b-2}$ is greater than $\dbinom{a-1}{b}$. This is where $b$ is equal to ...
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1answer
30 views

Interesting Combinatorics question relating the coefficients of variables in Pascal's Triangle

I tried this problem for a while by canceling the factorials on either side but for whatever reason, wasn't able to solve it. Could someone please help me? Is there a proof that ...
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2answers
18 views

creative method to obtain range of newton function ?!

I am searching for more proof that the range of $y=\frac{x}{x^2+1}$ is $ \frac{-1}{2}\leq y \leq \frac{+1}{2}$ these are my tries : domain is $\mathbb{R}$ first : $$y=\frac{x}{x^2+1}\\yx^2+y=x ...
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29 views

Proof Check In: Prove that $(\mathbb{Z}_n, +)$, the integers (mod $n$) under addition, is a group.

I received some help and direction on this from some users a few days ago, and have tried to take that information and craft it into something proofy. I would appreciate general suggestions, edits, ...
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3answers
46 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...
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2answers
54 views

Prove that if $a<1/a<b<1/b$ then $a<-1$

The following is Exercise 3.2.8 from Velleman: Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<1/a<b<1/b$ then $a<-1$. I solved it using the hint in the back of ...
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0answers
21 views

$x-y^4= LCM(x, y)$ [duplicate]

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 the least common multiple, but other than that, the textbook gave no hints ...
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1answer
44 views

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group.

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group. To show that this is a group, I know I need to show three things (in our text, we do not need to show that addition ...
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2answers
64 views

How can I be more confident that my proof is correct? (Real Analysis)

I am going through a textbook to prepare for Real Analysis and I recently tried the problem: Let $w\in\mathbb{R}$ be an irrational positive number. Set $A = \{ m+nw \mid m+nw > 0, ...
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2answers
57 views

Why is $f(x) = x^2$ uniformly continuous on [0,1] but not $\mathbb{R}$

According to How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity? There is a lot of agreement that $x^2$ is not uniformly continuous. But is $x^2$ uniformly ...
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1answer
21 views

Deriving the sum to product formula for sine using this method

I am trying to derive $sinC-sinD$ By this method: So far I have tried to set up the same method by beginning with $sin(A+B)-sin(A-B)$, but this reduces to a trivial zero and I can't find another ...
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2answers
58 views

Velleman's exercise $3.1.7$

Prove that if $a^3>a$ then $a^5>a$. Velleman gives this "hint": $$\text{One approach is to start by completing the following equation:}\ (a^5-a)=(a^3-a) \cdot x$$ I don't understand this ...
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4answers
85 views

Proving that the sum of the first $2n$ terms of the series $1^2 - 3^2 + 5^2 - \cdots$ is $-8n^2$ by induction

Use mathematical induction to prove the following for the first $2n$ terms of the series $$1^2 - 3^2 + 5^2 - 7^2 + \cdots = -8n^2.$$ As we have odd numbers that are squared we could use $n = ...
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1answer
43 views

Logical equivalence - Russell's Paradox

In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. ...
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1answer
27 views

Show that $T$ is the Set of All Sets Using the ZF Axioms

Let x be a set. Define the "set" $S = \left\{ y:x\subseteq y \right\}$ and $T = \cup\left\{y:y\in S \right\}$. Given any set $w$, let $z=x \cup \left\{w\right\}$. Then $x \subseteq z$, so $z \in S$. ...
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1answer
57 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
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2answers
57 views

Can someone explain to me why set proof involve the words “or” and “and”

For example, on proving the distributive law of set theory, the following constitutes as a proof Proof : I am new to proof involving sets but this to me seems nothing more than replacing unions ...
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1answer
28 views

Proofs of n^2 rem 4 [duplicate]

Show that if n is an integer than the remainder $(n^2 rem 4)$ = 1 or 0. I don't under what rem means in this form. Would it be n^2 + 4 = 1 or n^2 + 4 = 0?
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2answers
48 views

Prove that for any integer $m>1$, $\ \ (z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)]$.

Prove that for any integer $m>1$, $$(z+a)^{2m}-(z-a)^{2m}=4maz\prod\limits_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)].$$ This how tried to do it: Expand the two brackets on the right hand side ...
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2answers
23 views

Define a relation $D_n$ on $S$ by $xD_ny$ if and only if $x\mid y$. Determine if it's a poset.

Here is the question I am currently working on (screenshot): I'd appreciate some suggestions/guidance for part (a), proving that $D_n$ is a partial order. Reflexive: Let $x \in \mathbb{Z}$ ...
4
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1answer
34 views

Let $\ f_1:A \rightarrow B$ and $\ f_2:A \rightarrow B$. Prove or disprove $f_1 \cap f_2$ iff $f_1=f_2$.

Here is the question I am working on (screenshot): So, I haven't worked with function proofs very much (especially in the context of iff statements and with intersections). I am looking to see ...
2
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0answers
18 views

Applying rotation invariant linear operators to spherical harmonics

In the article "On boundary condition for multidimensional diffusion processes" A Venttsel says: I can't see how one can "prove that any other harmonic of order $n$ may be represented as a linear ...
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3answers
30 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
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3answers
54 views

What are the logical underpinnings of the epsilon- delta definiton of limits?

I'm having trouble getting my head around the epsilon-delta defintion of limits. I learned about conditional statements and I know that in order for a conditional to be true , one of the following ...
2
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2answers
32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
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2answers
39 views

Empty set Velleman's exercises

Doing an exercise from Velleman's 'How to prove it' I ended up thinking about exercise 2.3.8: Given that there are sets $ I=\{2,3\}, A_2=\{2,4\},A_3=\{3,6\},B_2=\{2,3\},B_3=\{3,4\}$. What is ...
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8answers
114 views

Prove by induction that $\frac{n^3}{3}+\frac{2n}{3}$ is an integer. [duplicate]

The question that I am working on is: Prove that $\dfrac{n^3}{3}+\dfrac{2n}{3} \in \mathbb Z \ \forall \ n \in \mathbb N$ The method that I think would be will work for this question is that I ...
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2answers
53 views

Induction question regarding Universe

I was given a question that looks like this. Prove that for each $2 \le n\in \mathbb N$, if $X_1,\ldots,X_n$ are subsets of some universe $U$, then the following is true: $$(X_1 \cup\cdots\cup ...
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2answers
47 views

How To Tackle Trigonometric Proofs involving $4$th and $6$th powers?

How do I prove that $\cos^4A - \sin^4A+1=2\cos^2A$ $\cos^6A + \sin^6A =1-3\sin^2A\cdot\cos^2A$ I was going through a very old and very rich book of Plane Trigonometry to build a nice foundation for ...
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3answers
68 views

Induction Proof using factorials

Recall that for $n \in N$, $n! = 1 \cdot 2 \cdots n$. Prove the following for each $n \in N$: $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$$ I ...
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4answers
71 views

Prove for each $n\in \mathbb{N}, 1^3 + 2^3 +\cdots+ n^3 = \frac{n^2(n+1)^2}{​4} ​ ​​​$ [duplicate]

So I was given a proof by induction question and here is my attempt $$1^3 + 2^3 + 3^3+\cdots+n^3= \frac{n^2(n+1)^2}{4}$$ $n=1$ $1=1$ Induction step: Assume statement is true for $n=k$, show true ...
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1answer
13 views

Prove that an underdetermined system of cannot have a unique solution(Is this proof correct?)

I know I misspelled underdetermine but is this proof correct? How can I improve it either way? Side Remark: Anyone who is down-voting please can you understand I new to this site and somewhat ...
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6answers
75 views

Prove that $xy+yz+zx \leq x^2+y^2+z^2$

Prove that $xy+yz+zx \leq x^2+y^2+z^2$ . Hint: Use $\frac{a+b}{2}\geq\sqrt{ab}$ First I tried using the hint by setting $a=x$ and $b=y+z$, however this results in the inequality: $$x^2+y^2+z^2 ...
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2answers
98 views

Understanding Spivak's alternative proof that $|a + b|\leq |a| + |b|$

For example, in Chapter 1 - Problem 14c Spivak asks the reader to come up with a different alternative proof that $$|a + b|\leq |a| + |b|$$ and this is what I found in the solution manual (with my ...
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3answers
57 views

proof by induction $2^n \leq 2^{n+1}-2^{n−1}-1$

My question is prove by induction for all $n\in\mathbb{N}$, $2^n \leq 2 ^{n+1}-2^{n−1}-1$ My proof $1+2+3+4+....+2^n \leq 2^{n+1}-2^{n−1}-1$ Assume $n=1$,$1 ≤ 2$ Induction step Assume statement ...
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4answers
44 views

A tautology that contains quantifier and logical connective.

It might seem a stupid "question", but I need a logical explanation of it. If $p(x)$ is a predicate and $q$ is a statement, then $(\forall x:p(x))\wedge q\iff \forall x:(p(x)\wedge q)$, and ...
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0answers
21 views

Is there any relationship between a worst matrix and its size and what are their common structures?

I am currently trying to test and calculate the worst possible $\mathcal{O}(f(n))$ for some algorithm. In order to do so, I need to find the worst possible (0,1) n x n matrix for some $n$s (e.g. ...
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0answers
12 views

boundary condition measure associated to a rotation invariant operator

According to A. Venttsel (On boundary condition for multidimensional diffusion processes) The measure in $(13)$ is of the form $\nu(drd\theta)\cdot d\varphi$ while in the general case we had ...
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2answers
59 views

Proofs by contrapositive. [closed]

Use proof by contrapositive to prove that the following statement is true for all integers $n$. If $3n^2+4n-5$ is odd, then $n$ is even.