For questions about the formulation of a proof, not about the mathematics behind it.

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0
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2answers
38 views

Proofs with for all statements including uniqueness and divides [closed]

Let $\mathcal{A}$ be a nonempty finite set of positive integers, with $\forall$ r $\in$ $\mathcal{A}$, $\forall$ s $\in$ $\mathcal{A}$ : r|s or s|r. (i). Prove $\exists$t $\in$ $\mathcal{A}$: t|a, $\...
1
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2answers
64 views

About $−\vert x \vert\le x \le \vert x \vert$ in absolute values

So, i'm really strugling with this one, when studying the triangle inequality, the inequality $−|x|≤x≤|x|$ pops really often, however I just can't get why this is true. I've seen only two different ...
3
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0answers
63 views

A different way to prove homeomorphism between rectangles and discs under standard $\Bbb R^2$ topology

I was reading about the classical problem to prove homeomorphism between subspaces of $\Bbb R^2$ a rectangle of the kind $R=\{(x,y):|x|\le a \land |y|\le b\}$ and some disc $D=\{(x,y): x^2 + y^2 \le r\...
1
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1answer
36 views

Banach Space Closed Subspace

Let $ \mathcal B$ be a Banach Space. Fix $z \in \mathcal B$ with $z \neq 0$. Consider the set $$A :=\{y-z : y \notin \operatorname{span} \{z\}, y \in \mathcal B\}.$$ Is it true that $\alpha z \notin ...
0
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1answer
29 views

Prove: $\exists !$ $t \in \mathbb{N}$ s.t. $\forall s \in \mathbb{N}$, $(t-9)s = 0$

I have a basic uniqueness proof to help me work on form: It should be obvious by simple inspection that the statement is true for t=9 and only for t=9. So my proof was this: Let $t=9$ then $9-9=0$, ...
1
vote
1answer
33 views

Proof of a six digit number [duplicate]

How can we prove that the number $142857$ is the only six digit number with the property that if I put the last digit before the first digit we get $5$ times our number so: $$\frac{714285}{142857}=5$$...
0
votes
1answer
40 views

Let $S$ be any set of statements. How do I concisely show that $\sim$ is reflexive, symmetric, and transitive on $S$?

The following problem is exercise 2.5.2 from "Mathematical Logic" by Ian Chiswell and Wilfrid Hodges (2007). I feel that the part about symmetry and transivity is a bit verbose and somewhat clumsy. ...
4
votes
1answer
147 views

Would like a hint for proving $(\forall x P(x)) \to A \Rightarrow \exists x( P(x) \to A)$ in graphical proof exercise on The Incredible Proof Machine

Update: Updated the title now that I've observed that we can use math in the title. I've also gone thru and removed dots. The tool expresses quantification using dots like this $\forall x.P(x)$ rather ...
1
vote
2answers
31 views

Find a family of open sets whose intersection is compact.

Does such intersection exists? im thinking about $An=(3+1/n;4+1/n)$ since $\bigcap An = [3,4] $ so its closed and bounded then its compact. Can someone please say whether its correct or not?
1
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0answers
32 views

Positivity of this improper integral

I am not relieved of the following problem and trip. Let $\varepsilon>0$ be a small parameter, $a>0$ be a given constant, $x_{\varepsilon}\in(0,a)$ be a given sequence such that $x_{\varepsilon}...
2
votes
1answer
26 views

Proofs problem with bijection [closed]

Let $f : A \rightarrow B$. Prove that if $X \subseteq A, Y \subseteq B$, and $f$ is a bijection, then $f(X) = Y$ if and only if $f^{-1}(Y) = X$.
4
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4answers
45 views

A surjective map $S \to T$ implies $|S| \geq |T|$

Problem: Suppose that there is a function mapping $S$ onto $T$. Show that $\operatorname{Card}(S)\ge\operatorname{Card}(T)$ Issue: I can't seem to find a reason why this follows. If $S$ maps $T$...
0
votes
1answer
122 views

Show that the set of polynomials with rational coefficients is countable.

Problem: Show that the set of polynomials with rational coefficients is countable. Idea: We know that the set of rational numbers is denumerable. This implies that the set of rational numbers is ...
1
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1answer
33 views

Square of an odd integer is odd, square of even integer is even, what is the case for higher powers?

Are there rules for higher powers? It seems like even and odd is preserved by powers, but how do I prove that?
1
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6answers
67 views

Induction Proof with Fibonacci

How do I prove this? For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.
1
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2answers
46 views

Prove that if the complex function $|f(z)|^2$ is constant in $D$ and $f(z)$ is analytic in $D$, then $f(z)$ is constant in $D$.

My proof: Let $|f(z)|^2 = M$ for $z\in D$. Then $f(z) = \pm\sqrt{M}$ (not sure about this step, are there only two values for the square root of a complex number> No right? Could be more. But I don'...
1
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1answer
32 views

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be functions. If $f$ onto and $g$ is not onto, then $g \cdot f:A \rightarrow C$ is not onto

I need help with this proof. I claim it is true, and I want to prove it directly using the definition of onto. Proof: Let $A,B,$ and $C$ be sets, and let f, g be functions s.t. $f:A \rightarrow B$ ...
1
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3answers
74 views

Proof using formal definition: Infinite limit

I was wondering how get the proof of this limit: $$\lim\limits_{x\to -\infty}\dfrac{{x^2} - x + 1}{x + 4} = -\infty$$ The problem is that I don't know what to do for find the appropriated values to ...
6
votes
4answers
922 views

Proof: Is there a line in the xy plane that goes through only rational coordinates?

Question: Is there a line in the XY plane that has all rational coordinates. Prove your answer. Idea: There is most certainly not. I believe it can be shown that between any 2 rational points that ...
8
votes
1answer
180 views

Prove that $\sqrt{2} + \sqrt[3]{3}$ is irrational [duplicate]

$\sqrt{2} + \sqrt[3]{3}$ is irrational ? These are my steps - $\sqrt{2} + \sqrt[3]{3} = a$ $3 = (a-\sqrt{2})^{3}$ $3 = a^{3} -3a^{2}\sqrt{2} + 6a -2\sqrt{2}$ $3a^{2}\sqrt{2}+2\sqrt{2} = a^{3}+6a-...
0
votes
1answer
39 views

How to negate this statement for a proof by contradiction

I want to try and construct a proof by contradiction but am having a hard time negating this statement. The statement that I am working with is There are only a finite number of points accepted ...
1
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2answers
63 views

Cardinality of all rational points in $R^3$

Question: Find the cardinality of the set of all points in $R^3$ all of whose coordinates are rational, and justify the answer. Idea: Call the set of all points in $R^3$ all of whose coordinates are ...
2
votes
1answer
42 views

cardinality of the set of points with one irrational coordinate, and one rational.

Find the cardinality of the set of all points in the plane which have one rational and one irrational coordinate. Justify you answer. My thoughts so far. We know that $\mathbb Q$, the set of all ...
0
votes
1answer
62 views

Proving Euler's formula using infinity sums

I want to prove $e^{i x} = \cos x + i \sin x$. Proof: $$e^{i x } = \sum \frac{x^{n} i^{n} }{n!} = -i\sum_{\textrm{odd}} (-1)^{n} \frac{x^{2n+1} }{(2n+1)!} + \sum_{\textrm{even}} (-1)^{n} \frac{x^{...
1
vote
1answer
53 views

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then…

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then which of the following is true $?$ $A)$ If $g$ is continuous then so is $f\circ g$ counterexample : $...
2
votes
1answer
47 views

Fibonacci identity $F_{n+1}^2 - (F_{n+1}F_n) - F_n^2 = (-1)^n$

I am trying to prove Let $F_n$ be the $n$th Fibonacci number. Then $F_{n+1}^2 - F_{n+1}F_n - F_n^2 = (-1)^n$ I am not sure where to start with this.
1
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1answer
56 views

Proof of arithmetic properties in $\mathbb{Z}$

In general, in elementary numbers theory when we prove properties we begin with natural numbers then I was wondering how you can extend the proof to $\mathbb{Z}$ clearly and properly. For instance, ...
2
votes
3answers
75 views

Is there a way of making “guess the next number in the sequence” rigorous?

This is maybe more of a question for matheducators.SE than math.SE but I'm more interested in the math than the education. A common problem given to middle and high school kids (at least in America) ...
-1
votes
1answer
60 views

Prove that if a relation R on a set A is reflexive, symmetric and antisymmetric, then $R=I_A$ [closed]

Prove that if a relation $R$ on a set $A$ is reflexive, symmetric and antisymmetric, then $R=I_A$ I know a relation is a set of ordered pairs and that $I_A = (x,x)$ but I have no idea how to do this ...
3
votes
1answer
74 views

Successful studying for a proof-based course final exam?

I'm currently taking a Transition to Advanced Mathematics course, which is entirely proof-based, so it's pretty new territory. Up until now, all the classes I've taken were fairly computational, so ...
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votes
4answers
56 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
1
vote
1answer
47 views

Difference between “Let x be a …” and “Let x … be arbitrary”

What is the difference between "Let x be a ..." and "Let x ... be arbitrary"? Consider the following example: Let $R$ be an equivalence Relation on $A$. Then $\forall x, y\in A.\ [x]_R=[y]_R.$ ...
1
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1answer
17 views

Let T(n) be defined by the following recurrence relation

Let T(n) be defined by the following recurrence relation $\begin{equation} \begin{cases} T(0) = T(1) = 1 \\ T(n) = T(n−1) + T(n−2) + 1 \quad for \quad n ≥ 2 \end{cases} \end{equation}$ Show that T(n)...
1
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0answers
17 views

Is this proof about the “Maximum Chunk Product Problem” (my own name for it) sufficient and clear?

- Question: For $L \in \mathbb{N}$, how do you choose $[x_i]$ so as to maximize $\prod x_i$ where $\sum x_i = L$ and $x_i \in \mathbb{N}$? (The number of "chunks" $x_i$ can be whatever you want.) - ...
1
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1answer
35 views

infimum of a sequence > 0, if the sequence converges proof

Let $(x_n)_{n=1}^{\infty}$ be a sequence of real numbers such that $x_n \neq 0$ for all $n \in \Bbb{N}$ and $x_n \rightarrow x$ as $n \rightarrow \infty$, where $x \neq 0$. Prove that $inf({|x_n|: n \...
2
votes
4answers
37 views

Geometry inequality proof

I started off with the given and by using the triangle inequality theorem but I don't know what to do next. Can someone please help? Thank you very much. I greatly appreciate it!
2
votes
1answer
13 views

How to Prove Triangle Centers in Tetrahedra

How would you prove the existence of triangle centers in tetrahedra, for example, the incenter, circumcenter, or centroid?
0
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1answer
40 views

What does it mean to 'show that' coefficients are a solution of this system of linear equations?

I don't really know what counts as a proof and haven't been taught maths since 16yo (29yo PhD now). I've got working knowledge of e.g. basic linear algebra, geometry, and statistics, but this feels ...
2
votes
4answers
119 views

(Proof) If $f$ and $g$ are continuous, then $\max\{f(x),g(x)\}$ is continuous

Consider the continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$. Show that $F:\mathbb{R}\rightarrow\mathbb{R}$ with $x\mapsto \max\{f(x),g(x)\}$ is continuous using the $\epsilon - \delta$ ...
0
votes
1answer
21 views

If $A$ is an open set of the taxicab metric $(R^2,d_1)$ then $A$ is a union of open balls of the Euclidian $(R^2,d)$ metric.)$

Is is true that if $A$ is an open set of the taxicab metric $(R^2,d_1)$ then $A$ is a union of open balls of the Euclidian $(R^2,d)$ metric. Where as taxicab metric I mean $R^2$ equipped with the ...
1
vote
1answer
47 views

Proof that $\|Ax\|<\|x\|$ for $A$ with absolute value of eigenvalues $<1$

I'm trying to prove that if all eigenvalues of the square matrix $A$ are $|\lambda|<1$ then $\|Ax\|≤\|x\|$. This seems like a simple proof but I fail to understand how to relate the basic ...
0
votes
2answers
58 views

The intersection of an infinite number of subspaces is a subspace

Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite ...
0
votes
1answer
68 views

Cardinality of set difference of finite sets

Is $|A \setminus B| = |A| - |A \cap B|$, where $A$ and $B$ are finite sets, true? I have been unable to prove this or find a good reference on cardinality of set differences. The only reference I ...
0
votes
0answers
35 views

problem from Stoll's introduction to real analysis [duplicate]

Suppose f: [a,b]$\rightarrow \Re $ is continuous. Let $M=max\{|f(x)| : x \in [a,b]\}$ Show that $$\lim_{n\rightarrow \infty}(\int_a^b |f(x)|^n)^{\frac{1}{n}}=M$$ My attempt: Suppose for ...
1
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2answers
47 views

How to prove that a number is composite?

How can one prove that a number is a composite number? I'm trying to prove that $6n + 1$ and $6n - 1$ are both composite for an infinite amount of integers $n$, with $n$ greater than or equal to $1$. ...
1
vote
3answers
65 views

Why must an inverse function be bijective?

Explain why $f^{-1}$ is a function if and only if $f$ is a bijective function. My attempt: $f^{1}$ is the inverse relation from B to A $\equiv$ function from B to A By definition of a function ...
1
vote
2answers
30 views

Ways to deal with generating sets of groups

I find that when I come across questions to do with generating sets of groups, I'm never quite sure how to go about the problem. It's difficult to deal with them purely set-theoretically, as you can't ...
-2
votes
1answer
43 views

How to write a proof of $ A\setminus B = \emptyset \leftrightarrow A \subseteq B$

I think the best way to prove this is by contradiction, but I'm struggling with the concept of how to write it properly. $$ A\setminus B = \emptyset \leftrightarrow A \subseteq B$$
0
votes
2answers
71 views

Proving $\mathbb{Z × N}$ is countable. [closed]

How would I prove that $\mathbb{Z × N}$ is countable? The hint given was to follow to indicated order. Thanks!
0
votes
0answers
17 views

Proving the treewidth of a graph is the maximum treewidth of the connected components

We have a graph $G = (V,E)$ and $C$ is the set of connected components of G. I want to prove that $tw(G) \geq $ Max $tw(C)$ where tw is the treewidth. I know the out sketch of the proof is to take ...