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

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5answers
72 views

Proof that $2^{2n}-1$ is not prime for $n \in \mathbb{N}, n > 1$

I notice that the number seems to be a multiple of 3: for n=2: $2^4 -1 = 15 $ for n=3: $2^6 -1 = 63$ for n=4: $2^8 -1 = 255$ How do I generalise?
1
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0answers
24 views

Proving concavity of a function

Let $f:[a,b]\rightarrow\mathbb{R}$ be twice-differentiable. Show that $f$ is concave if and only if $f''(x)\leq0$ for all $x\in[a,b]$. Moreover, if $f''<0$ for all $x\in[a,b]$, $f$ is strictly ...
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1answer
52 views

Need help understanding a proof about permutations in Rotman's ''Advanced Modern Algebra''

I need a hand in understanding the proof from "Advanced Modern Algebra" by Joseph J. Rotman of the following theorem: Let $\alpha \in S_n$ and let $\alpha = \beta_1...\beta_t$ be a complete ...
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1answer
35 views

Riemann Integration, question from Munkres

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2\rightarrow$ $\mathbb R$ be defined by setting $f(x,y) = 0$ if $x \not= y$ and $f(x,y) = 1$ if $x = y$. Show that $f$ is integrable over $[0,1]^2$ ...
0
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1answer
17 views

Well-formed formula a theorem in first-order system K

I'm trying to show that the following well formed formula is a theorem of K: $(\forall{x_i})\sim(A \to B) \to ((\forall{x_i})A \to (\forall{x_i})\sim B)$ Before going any further, I've rewritten it: ...
2
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2answers
42 views

Proof containing pairwise disjoint sets

I came across the following question while studying. Let $A,B,C,D$ be pairwise disjoint sets. Prove that if $|A| = |B|$ and $|C| = |D|$ then $|A \cup C| = |B \cup D|$. I thought of the fact that ...
8
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2answers
275 views

Axiom of Choice needed to “categorify” the cardinals?

I was playing around in $\mathsf{Set},$ trying to reduce it modulo isomorphisms to make a category $\mathsf{Card},$ letting the objects of $\mathsf{Card}$ be the isomorphism classes of $\mathsf{Set}$ ...
3
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1answer
60 views

Reading proofs vs. Attempting proofs, which one is more helpful?

In my discrete math course, I am often finding myself spending way too much time attempting a single math proof. I am starting to think that reading as many proof solutions as possible is a better ...
4
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1answer
35 views

Number of solutions to $|ax - bx| = a \;\text{or}\; b$?

While watching basketball tonight, I noticed that for 3, 4, and 6, $(6 \times 3) - (4 \times 3) = 18 - 12 = 6$. I thought this was a cool relationship and it led me to the following question: For ...
1
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4answers
62 views

If $0\lt y \le 1$, prove that there exists a unique positive real number x such that $x^2=y$

I'm stumped. I don't want an entire solution, just a hint. If $0\lt y \le 1$, prove that there exists a unique positive real number x such that $x^2=y$ The section in the book i'm on is the least ...
4
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2answers
102 views

Let $X$ an infinite $T_1$ space, then exist some subspace homeomorphic to $(\Bbb N,\tau)$ where $\tau$ is discrete or cofinite

My attempt to prove the statement of the title: if $X$ is infinite and $T_1$ with topology $\tau_1$ then any basis for $X$ is infinite too. If $X$ is $T_1$ then for any $x, y\in X$ with $x\ne y$ ...
1
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1answer
52 views

Proving that there exists a prime, p, between n! and n!+n iff p=n!+1

I have a conjecture (I believe this is not sophisticated enough to be a theorem) that there exists a prime, p, between n! and (n!+n) iff p=n!+1. I've tried a direct proof and a proof by contradiction ...
0
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1answer
36 views

Proving Inequality Statements From Proofs By Induction

Is it possible to prove $$1 + \frac{k}{2} + \frac{1}{2^{k+1}} \ge 1+ \frac{k+1}{2}\ \text{?}$$
0
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1answer
89 views

Proof of Euclid's Lemma in N that does not use GCD

I am looking for a proof of Euclid's Lemma, i.e if a prime number divides a product of two numbers then it must at least divide one of them. I am coding this proof in Coq, and i'm doing it over ...
3
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2answers
79 views

What is the function satisfying $\int_a^b f(x)\,dx =\int_a^b f'(x)\,dx$?

Let $$f:\mathbb R\rightarrow (0,\infty)$$ be a twice differentiable function such that $f(0)=1$, and further, for all $a, b \in \mathbb{R}$ such that $a < b$, $$\int_a^b f(x) \, dx =\int_a^b ...
2
votes
2answers
43 views

Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
2
votes
2answers
46 views

Delta epsilon argument in general

When I want to prove something in mathematics fe an expression goes to zero, I can either use basic rules of 'limits' or I can use the epsilon-delta method. I have a feeling that it's more consistent ...
1
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3answers
35 views

Need some hint about proving that if $X\cong Y$ and $X$ is $T_0$ then $Y$ is $T_0$

Im following the book Topology without tears of Morris (my first serious attempt to understand this area of modern maths as an amateur) and Im having a lot of trouble with proofs (proofs are a very ...
3
votes
1answer
46 views

$ \max \{ \underset{x \in X}{\sup } f(x)\ , \underset{x \in X}{\sup } g(x) \} = \underset{x \in X}{\sup } \{ \max \{ f(x), g(x) \} \}$?

Let $f,g: X \rightarrow [0, \infty )$ be bounded functions. Does hold that $$ \max \{ \underset{x \in X}{\sup } f(x)\ , \underset{x \in X}{\sup } g(x) \} = \underset{x \in X}{\sup } \{ \max ...
0
votes
2answers
32 views

Prove that all integers $n$ with $n \ge 12 $ is in the set.

$T = \{3k+7m: k,m \in \mathbb{Z}$, and $ k \ge 0, m\ge 0\}$ I assume I would do this by induction, so I want to show $3k+7m=n$ for $n\ge 12$. for the base case $n=12$. $3k + 7m = 12$ is true for ...
3
votes
3answers
134 views

prove if the function is vector space

Let $\Bbb R^3$ be 3-dimensional euclidean space. Is the set of all vectors in $\Bbb R^3$ that have the form $v=(v_1,v_2,v_3)$ where $5v_1-3v_2+2v_3=0$ itself a vector space? so my take: $v+w = ...
0
votes
0answers
32 views

Prove $n*2^n \leq 3^n$ using induction [duplicate]

I am trying to prove $n*2^n \leq 3^n$ for all $n \geq 1$ using induction. I tried to get it into the form $(n+1)*2^{n+1} \leq 3^{n+1}$ as follows: $n*2^n \leq 3^n$ $2n*2^n \leq 2*3^n$ $n*2^{n+1} ...
3
votes
3answers
69 views

Proving an inequality using mathematical induction

Using induciton, I have to prove following inequality: $$ 3^n > n2^n $$ I proved it for $n = 0$. Then assuming that the above is true, I try to prove it for $n+1$. So I start with: $$ (n+1)2^{n+1} ...
0
votes
0answers
56 views

Cyclic Group permutations

I am working on the following exercise question: Consider the following construction of a “keyed” hash function from Katz & Lindell (ex. 7.22 (1st ed.)/ 8.21(2nd ed.)). Gen : On input 1n , ...
0
votes
1answer
42 views

inclusion-exclusion principle proof (Without summations)

Suppose $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are sets with $\mathcal{A} \cap \mathcal{B} \cap \mathcal{C} = \emptyset$. Then $| \mathcal{A} \cup \mathcal{B} \cup \mathcal{C} |$ = ...
0
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2answers
35 views

Proofs with for all statements including uniqueness and divides

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
50 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 ...
1
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1answer
32 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
votes
1answer
28 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
32 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: ...
0
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1answer
36 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
123 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 ...
0
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2answers
29 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
30 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 ...
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 $f^{-1}(Y) = X$.
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4answers
34 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 ...
0
votes
1answer
43 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
17 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
53 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
36 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 ...
1
vote
1answer
26 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
vote
3answers
61 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
891 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
2answers
129 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} = ...
0
votes
1answer
38 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
vote
2answers
58 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
37 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
57 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} ...
1
vote
1answer
50 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 : ...