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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44 views

Proof using complete induction

Consider the set $S \subset \mathbb{N}^2$ of ordered pairs of integers defined by the following recursive definition: • $(3,2) \in S$ (basis) • If $(x,y) \in S$, then $(3x−2y,x) \in S$ (recursive ...
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0answers
46 views

Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
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1answer
42 views

Is there a proof for what I describe as the “recursive process of mathematical induction for testing divisibility”.

I was working on my homework for Discrete Math, and we were asked to "Prove: $6 | n^{3}+5n$,where $n\in \mathbb{N}$" my solution varied significantly from how I have seen it done by others. I noticed ...
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3answers
273 views

The correspondence theorem for groups

I had studied group theory a year ago, but still could not understand the proof involving The Correspondence theorem. let $G$ be a group and let $N⊴G$, where $N⊴G$ indicates that $N$ is a normal ...
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1answer
61 views

How to prove the binomial theorem

I'm trying to prove the following: $$ (p + q)^n = \sum_{x=1}^n \frac{n!}{x!(n-x)!}p^xq^{n-x}. $$ But I'm not sure where to start, would expanding the left hand side get me anywhere? Any tips or ...
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1answer
297 views

Use induction to prove that a function is not one to one

Suppose that m and n are positive integers with m > n and f is a function from $\{1, 2,\ldots, m\}$ to $\{1, 2, \ldots , n\}$. Use mathematical induction on the variable n to show that f is not ...
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1answer
62 views

“The Conjugate of a matrix”

I am having some trouble understanding a definition/question in my linear algebra text book. The question states " If $A$ is a square matrix, a matrix of the form $P^{-1}AP$ where $P$ is invertible ...
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2answers
153 views

Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$

I think I have the first part, as I have shown that $q < q + (q'-q)/2 < q'$, but I have trouble in proving that $|q-q'|< \epsilon$. Could someone tell me if there is a better way of showing ...
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2answers
51 views

Does there exist non-constant holomorphic $f: \mathbb{C} \to \mathbb{C}$ such that $e^{Re(f(z))}+ (\text{Im}f(z))^2 =1, \forall z \in \mathbb{C} \ ?$

Could anyone advise me how to prove/disprove that there exists non-constant holomorphic function $f: \mathbb{C} \to \mathbb{C}$ such that $e^{Re(f(z))}+ (\text{Im}f(z))^2 =1, \forall z \in \mathbb{C} ...
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1answer
19 views

Please help me finish this proof - the midpoints of the 4 sides of any quadrilateral are the vertices of a parallelogram

a) Let $A$ and $B$ be 2 points in the plane. Show that if $M$ is the midpoint of the line segment $\overline {AB}$, then $\vec{OM} = \frac{1}{2} (\vec{OA}+\vec{OB})$ where $O$ is the origin. I think ...
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1answer
67 views

Need help finding a proof strategy for a propositional logic theorem

Textbook is Ben-Ari's Mathematical Logic for Computer Science. This question is taken directly from the homework that my professor assigned, not from the textbook. Definitions of interpretations and ...
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1answer
38 views

Is the following claim true? (about a function that is strictly increasing and surjective)

We have a function $f$, $f:I \to F$, where $I$ is an open-ended interval, and $F$ is also an interval. $f$ is strictly increasing and surjective. I was trying to prove the fact that $f$ is continuous, ...
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1answer
62 views

Any function $f:\mathbb{R} \to \mathbb{R}$ is sum of two Darboux functions

From Wikipedia: Darboux functions are a quite general class of functions. It turns out that any real-valued function f on the real line can be written as the sum of two Darboux functions. This ...
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2answers
87 views

Fixed points of Holomorphic functions

Let $f:D(0,1) \to D(0,1)$ be a holomorphic map where $f$ has two distinct fixed points. Could anyone advise me how to prove $f$ is identity map? Do I use Schwarz-Pick somewhere? Let $g(z)=f(z)-z.$ ...
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0answers
61 views

Proof: At most 3 circles of radius 1/2 fit into the interior of a halfcircle of radius 1

It is a well known fact that at most 7 interior disjoint circles of radius 1/2 can be centered in a circle of radius 1; note that they don't need to be fully contained in the radius 1 circle. I am ...
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1answer
85 views

How to prove that two waveforms are rectangular (quotient zero)?

The problem is the following: Prove that, when Δf = integer multiple of $\frac{2}{T}$, the waveforms: $s_m(t) = A\cos(2πf_ct + 2πmΔft)$, for $m = 0, 1, ..., M - 1$ and $0 \le t \le T$ and $s_m(t) ...
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1answer
51 views

HWK Help: Conditional Probability Proof

So I've been working on this proof (and most likely making harder than it is) for quite some time now and I am getting nowhere. The proof is the following: Let $A_{k}$ be the event that the animal ...
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5answers
130 views

If $f=u+iv$ is an entire function such that $u^2\geq v^2,$ then $f$ is constant

Let $f=u+iv$ be an entire function such that $u^2(z) \geq v^2(z), \forall z \in \mathbb{C}.$ Could anyone advise me how to prove $f \equiv$ constant $?$ Hints will suffice. Thank you.
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1answer
36 views

Show that $f'_+(a)=f'(a+)$ if both quantities exist.

Show that $f'_+(a)=f'(a+)$ if both quantities exist. I'm not really sure where to start, any help is appreciated. I came up with this: If $f'(a^+)$ exists, then by definition $f'(a+) = \lim_{x\to ...
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2answers
74 views

On problems which can be proved easier if we use a different induction step.

Say we have a property $P$ defined on the natural nubers. Usually students are taught that to pove $P(n)$ is true for all $n\in\mathbb N$ you have to do the following: make a basis and use ...
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1answer
69 views

1-1 correspondence theorem

Here is the correspondence theorem stated as follows: Let $A$ be an Ideal of ring $R$.There is 1-1 correspondence between Ideals of $B$ containing $A$ and ideals of $R/A$. I have read the proof but ...
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1answer
48 views

if $A\subseteq \mathbb R^n$ is connected then is $A´$ (derived set) connected?

are the following statements true? 1) if $A\subseteq \mathbb R^n$ is connected then is $A´$ connected? 2)if $A´$ is connected then is $A$ connected? I can´t find any counterexamples. Can you help ...
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4answers
163 views

Show a sequence such that $\lim_{\ N \to \infty} \sum_{n=1}^{N} \lvert a_n-a_{n+1}\rvert< \infty$, is Cauchy

Attempt. Rewriting this we have, $$\sum_{n=1}^{\infty} \lvert a_n-a_{n+1}\rvert< \infty \,\,\,\Longrightarrow\,\,\, \exists N \in \mathbb{N}\ \ s.t,\ \ \sum_{n \geq N}^{\infty} \lvert ...
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1answer
264 views

Show that | and $\downarrow$ are the only binary connectives \$ such that {$} is functionally complete.

I've been reading and coping with van Dalen's Logic and Structure for a a few days. However, I've getting problems to solve his Exercise 6 from Ch 1 Sec 1.3 (p.28). In this exercise, van Daken asks ...
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5answers
2k views

Prime number between $n$ and $n!+1$

I am trying to prove that ($\forall \ n\in\mathbb{N}$) there exists a prime number $q$ such that $n < q \le 1 + n!$ I have made a graph with $n=0$ through $n=10$ and found solutions to all of them ...
3
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1answer
59 views

Show that $\log \left| z \right|$ is harmonic and find its the conjugate harmonic function.

Is the form correct for the conjugate harmonic? Attempt: First, we are given \begin{align*} \log \left| z \right| &= u(x,y) + iv(x,y) = \log \sqrt{x^2 + y^2} + i \cdot 0 \\ u(x,y) &= \log ...
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1answer
24 views

Prove that Autonomous are invariant under time translation

Reading my way through a big boy ODE book, and the authors write It is clear that if $\varphi(t)$ is a solution to $x'=f(x) \quad x(t_0)=x_0$, then clearly $\varphi(t+t_0)$ is a solution to $x'=f(x) ...
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2answers
66 views

Prove the existence of a greatest lower bound of $X$ if $X \subset \mathbb{R}$ is a non-empty set that is bounded below

Attempt: Let $C \subset \mathbb{R}$ be the set of all lower bounds of $X$. Since $C$ is not empty and bounded above, every $x \in X$ is an upper bound of every element $c \in C$. Thus, there exists ...
2
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1answer
41 views

Upper and/or lower Bound for Numbers of different topologies on the set $\{1,…n \}$

As the title says I am looking for upper and lower bound for the cardinality of different topologies on a set $\{1,....n\}$ for natural n! Are there some known bounds? My teacher says that there no ...
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1answer
97 views

How to Prove: If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.

I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go ...
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0answers
43 views

Proof Strategy: For all nonzero complex numbers $z$ and all nonzero rational numbers $a$ and $b, \mathbb Q (az+b)=\mathbb Q(z)$

I am having trouble proving (or finding a counterexample but I believe it to be true) the following. Prior to this I did some problems such as: Show that $\mathbb Q (-3+i\sqrt{2},2-\sqrt{8})=\mathbb Q ...
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2answers
25 views

If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$.

Here is a lemma whose proof is as under: If $S \in L(X,Y)$ and lim$_{r \to 0}\frac{\|Sr\|}{\|r\|}=0$,then $S=0$. Proof: The condition lim$_{r \to 0}\Big(\frac{\|Sr\|}{\|r\|}\Big)=0$means that ...
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2answers
52 views

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected Suppose $G$ is disconnected then $\exists A,B$ relatively open disjoint sets so that $A\neq ...
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4answers
69 views

If $f$ is 1-1, prove that $f(A\setminus B) = f(A)\setminus f(B)$

I'm having a tough time with this one. Here's the background: Let $X$ and $Y$ be sets, let $f:X\rightarrow Y$ and let $A,B\subseteq X$. For this proof, we also assume that $f$ is 1-1. I've already ...
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3answers
132 views

Proof that $\nabla(ab) = a\nabla b + b\nabla a$

I am trying to prove the following are equivalent: $a(x,y,z)$ and $b(x,y,z)$ $$\nabla(ab) = a\nabla b + b\nabla a$$ So looking at the left side: $\nabla(ab)= \cfrac{\partial ab}{\partial x} + ...
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1answer
30 views

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$

How to prove that if $x_n\to -\infty$ then $\frac{1}{x_n}\to 0$ as $n\to \infty$. My attempt: Let $x_n\to -\infty$ and $\epsilon\gt 0$. By the Archimedean Principle pick $N\in \mathbb N$ such that ...
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4answers
200 views

Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b\in \mathbb R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
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3answers
56 views

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational.

Find all $n\in \mathbb N$ such that $\sqrt{n+7}+\sqrt{n}$ is rational. By inspection it is pretty easy to see that the only $n$ that will work is $n=9$. Because the distance between perfect squares ...
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2answers
869 views

How to prove a function is not onto?

Let $f : Z\to Z$ be the function defined by $f(x) = 3x + 1$. Prove that $f $ is not onto, using a proof by contradiction. (Choose an integer $n$, and then prove ($\forall m \in Z$)($f(m) ≠ n$) by ...
2
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0answers
27 views

Given the sets $X$ and $Y$ in the real numbers with least upper bounds $a$ and $b$ respectively, prove that $a+b$ is the least upper bound for $X + Y$

I've seen this proof done other ways and I wonder if my way is right. It's very similar to the $\epsilon > 0$ approach I've seen elsewhere but uses a contradiction: Let $X$ and $Y$ be sets of real ...
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2answers
286 views

The boundary of the union of two sets is a subset of the union of boundaries

I'm stuck on trying to get this proof started. I want to prove that $\delta(S_1 \cup S_2)\subset \delta S_1\cup\delta S_2$, where $S$ is some set. I don't need a full proof, just a hint to get ...
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2answers
37 views

induction for idempotent matrix : $P^n = P$

Given that $P^2 = P$ how do i prove by induction that $P^n = P$? I have tried the following: we know that $P^k = P$ holds for $k = \{1,2\}$. If we now take $k=3$: $$ \begin{align} P^3 &= ...
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3answers
244 views

prove this inequality with log and positive value “x”

How do I prove that for every positive $x$ , $1-x \le -\log{x}$ Can I use convexity somehow?
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1answer
172 views

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ [duplicate]

Prove ${2n\choose n}=\sum\limits_{k=0}^n {n\choose k}^2$ My Approach: I will be making use of $$\tag 1\quad{m+n\choose r} = {m\choose 0}{n \choose r} + {m\choose 1}{n\choose r- 1} + ...
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1answer
163 views

Practice Examples of Proofs by Induction, Direct/Indirect Method

I'm learning about proofs in school, quite a few different sorts (but not geometry ones), but the teacher is teaching by slides mainly, not books. The main ones are proof by ...
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2answers
81 views

Induction and Maximum Principle

I wish to show that the following two assertions are equivalent: (Principle of Mathematical Induction) Let $S$ be a nonempty subset of the set of non-negative integers satisfying the following two ...
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1answer
53 views

For any two Ideals $A$ and $B$,$A+B=\langle A \cup B \rangle$

Below is the proof of : Prove that for any two ideals $A$ and $B$ of ring $R$,$A+B=\langle A \cup B~\rangle$ . Proof: By theorem (for any two ideals of a ring $R$ ,then the set $A+B$ is an ...
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1answer
258 views

When to use weak, strong, or structural induction?

For weak induction, we are wanting to show that a discrete parameter n holds for some property P such that P(n) implies P(n+1). For strong induction, we are wanting to show that a discrete parameter ...
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1answer
88 views

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$

Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$ My Approach Let $x_k$ be one element in a set of $n$ elements. $n-1\choose r-1$ $=$ the number of unique groups of $r$ containing ...
0
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1answer
35 views

Proof with Cartesian coordinates.

Let $S_b := \{(x,y) \in\mathbb R^2 | y = 3x + b\}$ where $b\in\mathbb R$. Give a direct proof that if $(r,s)\in\mathbb R^2$, then there exists a $b\in\mathbb R$ such that $(r,s) \in S_b$. I have ...