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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1
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4answers
189 views

How do I write this proof formally?

How can I formally prove that $$\max\{\lvert x+y\rvert _i \} \leq \max\{ \lvert x_j \rvert \} + \max\{\lvert y_k \rvert \}$$ Where $x,y$ are the components of a $n$-vector with $1 \leq i,j,k \leq ...
3
votes
1answer
38 views

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring $R$, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with. Please be ...
0
votes
2answers
32 views

Power set equinumerosity. Is this proof correct?

So I'm trying to prove the following, Prove that if $A\sim B$ then $\mathscr{P}(A) \sim \mathscr{P}(B)$. Here's how I started out to prove there is a function that is injective: Suppose $A ...
4
votes
1answer
36 views

Introduction to proofs. [duplicate]

I am not at all familiar with mathematical proof-writing and would like to learn how to create my own proofs. So, I was wondering whether it would be possible for you to recommend me to any book or ...
1
vote
2answers
41 views

Summation Proof Dealing With 3s Multiples [duplicate]

So the problem is as follows: Prove that if the sum of digits of a decimal $n$ is three's multiple, then n is three's multiple by direct proof. For example, $11234567$ is 3's multiple because ...
-2
votes
0answers
47 views

Summation Direct Proof Help [on hold]

Prove that if the sum of digits of a decimal n is three's multiple, then n is three's multiple by direct proof. For example, 11234567 is 3's multiple because 1+1+2+3+4+5+6+7=24, and in fact, 11234567 ...
0
votes
1answer
33 views

How to prove this theorem? (Logical symbols help)

This is a theorem from a book. I'm having a hard time on proving it. Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of ...
1
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2answers
35 views

Proving $10^n \equiv 1 \pmod 3$ for all $n\geq 1$ by induction

Prove that $10^n \equiv 1 \pmod 3$ for all positive integers $n$ by mathematical induction. Can someone please help me in solving this problem and explain what's going on? Any guidance would be ...
1
vote
1answer
31 views

Formal language: Proving the reverse operation on a word through induction

I'm practicing proofs and given the following statement: Let $\Sigma$ be an alphabet, $\epsilon$ the empty word and $\sigma:\Sigma^{*}\rightarrow\Sigma^{*}$ an operation which for $a\in\Sigma$ and ...
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0answers
38 views

Why does this proof by bashing not work?

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. The feet of the perpendiculars from $C$, $B$, and $A$ to the opposite sides are $F$, $E$, and $D$ respectively. Prove that ...
1
vote
1answer
57 views

Proving that the gamma function is a certain limit

This time I want to prove that $\displaystyle \Gamma(x) = \lim_{\varepsilon\rightarrow 0} \int_\varepsilon ^{1/\varepsilon} t^{x-1} e^{-t}$, I know this is true because we have defined $\displaystyle ...
0
votes
1answer
13 views

Using existential instantiation on a universally quantified given

I'm trying to prove the following exercise of How to Prove it: A structured Approach (Section 3.4, exercise 19): Suppose A, B and C are sets. Prove that A $\triangle$ B and C are disjoint iff A ...
1
vote
0answers
18 views

Is there a standard notation for $(p_i-k)(p_{i-1}-k)(p_{i-2}-k)\cdots$ where $k$ is a small positive integer

For $k=0$, there is: $p_i\# = (p_i)(p_{i-1})(p_{i-2})\cdots(5)(3)(2)$ For $k=1$, there is: $\varphi(p_i\#) = (p_i-1)(p_{i-1}-1)(p_{i-2}-1)\cdots(5-1)(3-1)(2-1)$ Is there any other notation that ...
3
votes
1answer
14 views

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
2
votes
0answers
19 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
44 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
votes
0answers
54 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
votes
0answers
12 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
votes
0answers
30 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
24 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
32 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
246 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
93 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
24 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
52 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
149 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
72 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
43 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
32 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
60 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
0answers
47 views

Proving Limits as n->Infinity (Intro to Analysis) [closed]

I'm having a really tough time learning how to prove a limit as n goes to infinity. The book I have gives no examples and my professor certainly hasn't made anything clearer as of yet. If anyone could ...
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
76 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
45 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
30 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
25 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
17 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
41 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
140 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: ...