For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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2
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0answers
23 views

Proof: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$

Prove: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$ This is my work so far: Let P be any point of the plane and set: $P'=T_{AB} (P)$ We want to show ...
0
votes
2answers
49 views

Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
0
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0answers
55 views

Prove that $\int_{a}^{b} f(x)g'(x) dx = 0$ iff $f$ is constant

Given that $f$ is continuously differentiable and increasing on $[a, b]$, $g$ is differentiable on $[a, b]$, and $g'$ integrable on $[a, b]$. If $g$ is positive and $g(a) = g(b) = 0$, show that ...
0
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0answers
13 views

Upper bounds and Lower bounds (Relations Proof Problem)

So I've only recently started studying proofs and I've been using Velleman's "How to Prove it" This is a theorem from the book. I'm having a hard time on proving it. Suppose A is a ...
0
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0answers
36 views

Prob. 2.7-10 in Kreyszig's Functional Analysis Book: Is my solution good enough for anciliary purposes?

With valuable help from the SE community, I've managed to come up with the following solution to Prob. 10 after Sec. 2.7 in Introductory Functional Analysis With Applications by Erwine Kreyszig. I ...
0
votes
1answer
26 views

Prove that the set $[0,1)$ is a closed set in the half-open interval topology of $\mathbb{R}$.

Prove that the set $[0,1)$ is a closed set in the half-open interval topology of R. I know that I need to show that the complement of this set is open in order to show that this set is closed. The ...
0
votes
2answers
51 views

Prove or disprove: There exists a prime p > 3 such that p + 2 and p + 4 are also prime

I'm having a lot of difficulties with this proof. Can someone please solve it and explain to me what's going on at each step? Thank you!
1
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2answers
34 views

Proof by Elements to Show $D^{c} ⊆ A^{c}$

Use proof by elements to verify that for all nonempty sets $A$, $B$, and $D$ if $A ⊆ B$, $D^{c} ⊆ B^{c}$, then $D^{c} ⊆ A^{c}$. Here's the proof I have written so far. I have gotten feedback that ...
2
votes
2answers
251 views

how to prove that the following is not a regular language?

the language we want to disprove is : $$ L = \{ 0^i1^j| gcd(i,j)=1 \} $$ my attempt : i used the pumping lemma this way: consider the set of strings of the form $0^p1^q$ such that $n <=p$ and ...
4
votes
2answers
96 views

How much does Proof writing improve over the years?

This is a very soft question. Just a bit of background: I'm a junior in high school taking Analysis I and II out of Baby Rudin at a very well-recognized university. I find quite a few of his ...
0
votes
1answer
26 views

Let S ⊂ R 2 . Show that if cl(S) is convex, then S is convex.

Let $S ⊂ R^2$. Show that if $cl(S)$ is convex, then S is convex. This seems intuitive but, I am having trouble thinking of a proof or counter example.
0
votes
0answers
53 views

Proving Integrability of $sgn(\sin(\frac{\pi}{x}))$

I must show that for $f(x) = sgn(\sin(\frac{\pi}{x}))$ on $[0,1]$, that $f$ is Integrable. I know that a function is integrable if the Upper and Lower sums of $f$ coincide. That is, if $$U(P,f) - ...
0
votes
2answers
29 views

Proving that the set of languages over an alphabet Σ is a monoid regarding concatenation

I'm practicing proofs and would like to prove that the set of languages over an alphabet $\Sigma$ is a monoid regarding concatenation by showing that the following statements are true: There is a ...
0
votes
2answers
156 views

Proving $x^{2}+1 \neq n! $,using Gaussian Integer.

I want to show that $$x^{2}+1 \neq n! $$ for $n>3$ where $x,n$ are both integers. Since $$x^{2}+1=(x+i)(x-i) $$ it follows that $x^{2}+1$ has only prime factors on the form ($4k+1$), whereas $n!$ ...
1
vote
4answers
75 views

Suppose that $F'(x)\leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x)\leq G(x)$ for all $x \in \mathbb{R}$.

Suppose that $F'(x) \leq G'(x)$ for all $x \in \mathbb{R}$. Then $F(x) \leq G(x)$ for all $x \in \mathbb{R}$. Prove or disprove. I came up with counterexample that $\cos(x) \leq \cos(x)+1 $ for all ...
1
vote
3answers
44 views

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length.

The curve segments $y=e^x$ for $0\leq x \leq 1$ and $y = \ln(x)$ for $1 \leq x \leq e$ have the same length. Prove or disprove. I got the idea that they are inverse functions and probably we can show ...
1
vote
3answers
33 views

How to prove that $G(x)=ax+b$ is a one-to-one correspondence where $a\neq0$ and $a,b\in\mathbb{R}$.

$G(x) = ax + b$ where $a$ is not equal to $0$ and $b$ are real numbers. Prove $G$ is a one-to-one correspondence. I understand that for every $a$ there is a corresponding $b$-value that does not ...
1
vote
3answers
61 views

Prove that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$

I have been tasked with proving that $(n!)^{\frac{1}{n}} < ((n+1)!)^{\frac{1}{n+1}}$ for every integer $n \geq 1$. My instinct is to use induction, but I have gotten stuck. Base Case - $n=1$: ...
0
votes
2answers
49 views

How to write the below proof rigorously for : If $p$ is a boundary point of $S$, then $p$ is a lim point of $S$

How to write the below proof rigorously for : If $p$ is a boundary point of $S$, then $p$ is a lim point of $S$? We have: $p$ is a boundary point of $S$ means that $$\forall r\gt 0, \exists a \in ...
0
votes
1answer
51 views

Dedekind Cuts and Real Numbers

A Dedekind cut L is a nonempty proper subset of the rational numbers that: (1) Has no maximal element (2) for all a,b in the rational numbers a is in L and b < a implies that b is in L. If $D$ is ...
3
votes
1answer
47 views

Prove that $R$ is an equivalence on $\mathscr P(A)$. Is this correct?

Suppose $B\subseteq A$, and define a relation R on $\mathscr{P}(A)$ as follows: $$R=\{(X,Y)\in\mathscr{P}(A) \times \mathscr{P}(A)\mid(X\mathrel{\triangle} Y)\subseteq B\}$$ Prove that $R$ is an ...
1
vote
4answers
78 views

Prove that if $3|mn$, then $3|m$ or $3|n$

I am trying to prove this for integers $m$ and $n$. I tried to reach prove that $3|m$ by assuming that 3 does not divide $n$, but this is such a basic assumption of mine already that it is hard for ...
1
vote
1answer
47 views

Proof by cases: Prove that if $x$ and $y$ belong to the set of real numbers, then $\max(x, y) + \min(x, y) = x + y$

Question: Let $x$ and $y$ be real numbers. Using a proof by cases, show that $$\max(x, y) + \min(x, y) = x + y.$$ So for this question, I'm not sure how you would apply proof by cases. I think that ...
1
vote
0answers
27 views

Regularity of Special Measures

(1) Show that the counting measure on $\Bbb Z$ with the induced metric from $\Bbb R$ is regular. (2) Show that the delta measure with respect to a point $x_0$ on any metric space is regular. What I ...
1
vote
2answers
31 views

Proving intervals are equinumerous to $\mathbb R$

Let $a$, $b$ elements of $\mathbb R$ with $a < b$. By combining the results of the past exercises and examples, show that each of the following intervals are equinumerous to the set $\mathbb R$ ...
0
votes
1answer
39 views

Proving intervals are equinumerous

a.) Show that (0, 1] is equinermous to the interval (0, 1) by giving an example of a bijection from (0, 1] to (0, 1). My attempt: ...
0
votes
1answer
33 views

Write an inductive proof that if there is a surjection $ f : \lceil m \rceil → \lceil n \rceil $ then $m ≥ n$.

Here's the problem: Write an inductive proof that if there is a surjection $ f : \lceil m \rceil → \lceil n \rceil $ then $m ≥ n$. Where I Am: I assume that I should induct on $n$ and come to the ...
3
votes
1answer
26 views

Prove: R∩R−1 is symmetric.

The problem that I'm having is proving it - obviously. The only context that I am provided with is: "Prove: R∩R−1 is symmetric." If (x,y) ∈ R then (y,x) ∈ R−1, and since it's the intersection, ...
0
votes
1answer
33 views

Prove that there is a surjection $ f:X \to Y$ if and only if $ |Y| \le |X| $.

Here's the problem: Let $X$ and $Y$ be sets. Prove that there is a surjection $$ f:X \to Y$$ if and only if $$ |Y| \le |X| $$. My work so far: I am working on the following direction: If $ |Y| ...
0
votes
0answers
21 views

combinatorial proof of Vandermonde's Identity [duplicate]

So I can not figure out the combinatorial proof for Vandermonde's Identity for the example $\sum_{i=0}^k \binom {k} {i}^2 = \binom {2k} {k}$ Any help would be appreciated. Figured it out, thanks :)
2
votes
4answers
72 views

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $ 7\mid b$

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $7\mid b$ What I did: I found the possible remainders for $a^2$ are $0, 1, 2$ and $4$. I think I should say $r_7(a^2)+r_7(b^2)$ can't equal any ...
1
vote
1answer
42 views

Prove that S is an equivalence relation on P(X)

Let $X = \{ 1, 2, 3, \ldots , 2015\}$ and $Y = \{ 1, 2, 3,\ldots, 271 \}$. Let S be the relation on power set (X) defined by For all A, B in $\mathcal{P} (X)$, $(A, B) \in S$ if and only if $|A \cap ...
0
votes
1answer
20 views

generalised eigenspace is invariant proof

let $T \in \text{End}(V)$ where $V$ is a finite dimension vector space. We define $V_j(\lambda) = \ker ((\lambda I - T)^j))$ as the generalised eigenspace. I am trying to prove that this space of is ...
2
votes
1answer
40 views

Every vector space has a basis

Prove that every vector space has a basis. I am going to use Zorn's lemma for this also here is a necessary definition regarding totally ordered subsets: one element will be contained in the other. ...
3
votes
5answers
143 views

Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$

The problem i have is: Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$ Ive been trying to use different examples of similar problems like at: ...
0
votes
2answers
22 views

Short proofs about integrability

If true, the prove it; if false, the provide a counterexample. a) If $f$ is integrable, but $g$ isn't, then $f + g$ is not integrable. True: Assume that $f + g$ is integrable, then $f$ and $g$ must ...
0
votes
1answer
50 views

Probability proof and graphs

Let $G = (V, E)$ be a graph with vertex set $V$ and edge set $E$. A subset $I$ of $V$ is called an independent set if for any two distinct vertices $u$ and $v$ in $I$, $(u, v)$ is not an edge in $E$. ...
1
vote
2answers
43 views

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$ "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss ...
0
votes
0answers
82 views

Edge and Vertex set proof using an algorithm

Disclaimer: This is a homework question, so no direct answers please. All that I'm looking for is a good springboard to get started from with this question, as it has been tearing me apart for the ...
0
votes
0answers
32 views

Existence of non-coprime between an integer and an arithmetic sequence

Take two relatively prime numbers $m,n \in \mathbb{Z}$ (i.e. $gcd(m,n) = 1$) where $m \neq 1$. Show that: $$\forall a \in \mathbb{Z} \textrm{ s.t. } 0<a<n$$ $$\exists i \in \mathbb{Z^+} ...
-1
votes
5answers
84 views

Prove that the sequence $\sin\left(\frac{n\pi}{3}\right)$ diverges

I don't want to hear that since $sin$ is a periodic function, etc, then we are done. I would like to see a simple proof that make use of the definition of convergence of a sequence. I have tried to ...
1
vote
1answer
23 views

Examples on how to give a proof or a counterexample of a statement

Examples; Prove or give a counterexample of the following statements,with quantifiers: 1) For each non-negative number s, there exists a non-negative number t such that s≥t 2) For each non-negative ...
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2answers
27 views

how to prove the uniqueness and existence of equations

I've the equation $e^x=5$, know it has the solution $x=\ln 5$. How to prove the existence before, and after the uniqueness of this solution?
1
vote
3answers
16 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
0
votes
2answers
47 views

If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$

Just wanted to confirm that this is a correct solution: Proof: Suppose $f(x_0) > 0$ for some $x_0 \in [a,b]$. Then, by continuity of $f$, for $\epsilon < f(x_0)$, there exists $\delta > 0$ ...
-1
votes
5answers
58 views

Discrete Math proving gcd [closed]

I have a proof that states: Let $n$ be an odd natural number. Prove $\text{gcd}(n,n+1)=1$. I have no idea where to begin, any advice?
4
votes
5answers
265 views

Show that $g(a) = g(b) = 0,\ \int_a^b f(x)g(x)dx=0 $ implies $f(x)=0$

Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, I want to ...
2
votes
2answers
37 views

Lebesgue outer Measure of a face of rectangle in $\Bbb R^{n}$

Show that the outer measure of a face $I_1 \times \dots \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$ of a rectangle $I_1 \times \dots \times I_n \subset \Bbb R^{n}$ is zero. ...
-1
votes
2answers
67 views

Identity exists [closed]

I am stuck on this proof. Not sure where to begin. Any help would be appreciated
1
vote
4answers
62 views

Prove $(2n + 1) + (2n + 3) + \cdots + (4n - 1) = 3n^2$ by induction

This might be an easy problem for you, but I am having difficulties in understanding the formula. As we can see, we have a pattern $$2n + \text{odd number}$$ in $$(2n + 1) + (2n + 3) + \cdots + ...