For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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0
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1answer
36 views

Proof Check: The cardinality of the set of all binary series with an infinite amount of 0's and 1's:

Label the set of all binary series with an infinite amount of 0's and 1's as $C$. It's easy to prove that the set (labeled $A$) of all binary series with a finite number of 1's is countable. I can ...
2
votes
0answers
115 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
1
vote
1answer
79 views

Proof Check: The cardinality of all monotonically increasing series of natural numbers.

Given an infinite series, $a_1a_2a_3\ldots$ define $F$, $F(a_1a_2a_3\ldots) = a_1\bmod(2)a_2\bmod(2)a_3\bmod(2)\ldots$ It's trivial to show that $F$ is onto the set of all infinite binary series, ...
3
votes
1answer
355 views

Inverse image of $\sigma$-algebra

Is this proof correct? Let $f$ be a function mapping $\Omega$ to $E$ with $\mathcal E$ a $\sigma$-algebra on $E$. Show that $\mathcal A=\{f^{-1}(B):B\in \mathcal E\}$ is a $\sigma$-algebra on ...
2
votes
2answers
53 views

Sub $\sigma$-algebra

Is my proof of the following correct? Let $\mathcal{A}$ be a $\sigma$-algebra on $\Omega$ and let $B\in\mathcal{A}$; then $\mathcal{B}=\{A\cap B:A\in\mathcal{A}\}$ is a $\sigma$-algebra on $B$ ...
5
votes
1answer
106 views

Endomorphisms of forgetful functor $\mathbf{Grp}\to \mathbf{Set}$

It is well known endomorphisms of faithful functor form a monoid. I was trying to determine monoid of endomorphisms of forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, and found it to be ...
2
votes
5answers
100 views

Prove that $A \ne B$ is equivalent to the logical statement $(\exists x)[x \in A \land x \notin B] \lor (\exists x)[x \in B \land x \notin A]$

Prove that $A \ne B$ is equivalent to the logical statement $(\exists x)[x \in A \land x \notin B] \lor (\exists x)[x \in B \land x \notin A]$. Given: P: $A \ne B$ is equivalent Q: the logical ...
2
votes
2answers
57 views

Let $(G,\cdot)$ be a group and $a\in G$. Define $x\ast y:=x\cdot a \cdot y$. Show $(G,\ast)$ is a group. [duplicate]

Let $G$ be a group with operation $\cdot$ and let $a \in G$. Define a new operation $*$ on the set $G$ by $x*y$ = $x·a·y$ for all $x,y \in G$. Show $G$ is a group under the operation $*$. Does this ...
0
votes
0answers
60 views

Proof verification and suggestion to elude the AC (equivalent definition of adherent points).

Hi everyone I'd like to know if the following is correct and, more importantly, if there is some way to escape of the axiom of choice (as the hint the book says "use AC"). Definition: Let $X\subset ...
4
votes
0answers
554 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
5
votes
0answers
232 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
0
votes
6answers
92 views

Prove the following statement: If $E$ is an empty set and $A \subseteq E$, then $A$ is an empty set.

If $E$ is an empty set and $A \subseteq E$, then $A$ is an empty set. Edit: Thanks for the \emptyset Latex command. Given: P: $E$ is an empty set and $A \subseteq E$ Q: $A$ is an empty set. We ...
2
votes
1answer
81 views

Prove the following statement: If A is any set, then $A \subseteq A$

I'm doing some practice problems and I'm wondering if I got this right. I think this is a very short proof, but I'm not sure. Given: P: A is any set Q: $A \subseteq A$ We have a $P \rightarrow Q$ ...
5
votes
2answers
86 views

$f$ integrable $\Leftrightarrow f<\infty$ a.s.?

$f\colon\Omega\to\mathbb{R}$ measurable function on measure space$(\Omega,\mathfrak{A},\mu)$. I am interested to know if then $$ f\text{ is integrable }\Leftrightarrow f\text{ is finite a.s.}~~~. $$ ...
1
vote
2answers
109 views

Elementary properties of closure

Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks. ...
2
votes
3answers
83 views

Simple closed set proof

I want to show that $$\{ x: x_1 +x _2 = 1, x_1, x_2 \geq 0\}$$ is closed. Let $(x_1^*, x_2^*)$ be limit points of the set with $(x_1^{(k)},x_2^{(k)})\rightarrow (x_1^*, x_2^*)$ where the ...
1
vote
2answers
88 views

Wronskian determinant and Linear dependence

I was trying to show that if functions f and g defined on interval I are linearly dependent then the Wronskian determinant is zero. Suppose f, g $\in$ I and f g are linearly dependent, then $\forall ...
3
votes
1answer
96 views

Determining and proving supremum and infimum of given set.

Let $ A\subset\mathbb{R} $ a bounded set, $ \mathbb{N} = \{0,1,2,...\} $ and $$ B:= \left\{\frac{n}{n+1} - a \mid a \in A, n \in \mathbb{N}\right\}$$ Give inf B and sup B in terms of inf A and sup ...
0
votes
2answers
113 views

Combinatorics - Check my answer, sitting order, round table.

$n$ people are sitting at a round table with $n$ seats at a restaurant. The restaurant has only 2 dishes, steak and salad. How many ways are there for the diners to choose a dish, such that no 2 ...
2
votes
1answer
3k views

Bonferroni inequality proof

Is this proof for $P(\bigcup_{i=1}^n A_i)\le\sum_{i=1}^nP(A_i)$ correct? Pf. By induction. For $n=2$, $$P(A\cup B)=P(A)+P(B)-P(A\cap B)\le P(A)+P(B)$$ Assume that the statement is true for $n-1$, ...
3
votes
1answer
289 views

Prove Intersection of $\sigma$-algebras is a $\sigma$-algebra and the powerset is a $\sigma$-algebra

Fix a set $\Omega$. A $\sigma$-algebra on $\Omega$ is a non-empty collection of subsets of $\Omega$ closed under taking complements and countable unions. I'd like to prove that (1) for finite ...
2
votes
2answers
91 views

Are the proofs I made correct?

Edit: Since these are pretty small assignments each and all of the same topic, I've decided to post them into one thread. I hope that's ok. Thank you. Question I have the following assignment: ...
0
votes
2answers
390 views

Using proof of equivalence

I just wanted to make sure whether I was on the right track or not with this. Let $r\in\mathbb{R}_{\ne0}$. Use a proof of equivalence to show the following: $$r\in\mathbb{Q} \iff ...
2
votes
1answer
152 views

Modified Euclid's proof of infinite primes

Q. Alternate the proof for Euclid's infinite number of primes to show there are infinitely prime numbers of the form $6n-1$ where n is an integer. my attempt, suppose by contradiction there are ...
0
votes
0answers
65 views

Showing finite additivity of Lebesgue measure

I want to show that Lebesgue measure is finitely additive on the set of semi open rectangles of the form $[a,b)$ here is what I did ($\sqcup$ is disjoint union , $\lambda^n$ is n-dimensional Lebesgue ...
2
votes
3answers
136 views

If 1 $\leq x$, then $\sqrt{x} \leq x $

This is a really simple problem but I am unsure if I have proved it properly. By contradiction: Suppose that $x \geq 1$ and $x< \sqrt{x}$. Then $x\cdot x \geq x \cdot 1$ and $x^2 < x$ ...
3
votes
0answers
99 views

Monotonicity of measures

Let $\mu$ be a measure defined on $\Omega$. Then $\mu(A)\le \mu(B)$ for all $A\subset B\subset \Omega$. pf. Let $A\subset B$, let $C=A^c\cap B$. Then $A\cap C=\emptyset$ and $A\cup C = B$. By ...
1
vote
1answer
87 views

Proving $\phi(x) = \psi(x)$ for all $x \in G$ where $\phi: G \rightarrow G'$ and $\psi: G \rightarrow G'$ and $G$ and $G'$ are isomoprhic.

Consider the following task: Let $G$ be a cyclic group with generator $a$, and let $G'$ be a group isomorphic to $G$. If $\phi: G \rightarrow G'$ is an isomorphism, show that, for every $x \in G$, ...
1
vote
2answers
133 views

Showing 3 is an irreducible element of $\mathbb Z[\sqrt{2}]$

What I tried: $$(3)=\{3r|r\in \mathbb Z\}\space\mbox{is a maximal ideal of}\space\mathbb Z\implies(3)=\{3(a+b\sqrt{2})|a,b\in\mathbb Z\}\space\mbox{is a maximal ideal of }\mathbb ...
3
votes
2answers
108 views

Proof of integral

Is there an analytical method to show that $$ \int_{-a}^a\exp\left(\frac{-1}{1-(x/a)^2}\right)\,\mathrm{d}x=ka, $$ for $a>0$. I have confirmed this result numerically for a range of values of $a$. ...
1
vote
1answer
96 views

If $L_1 \cap L_2$ is decidable, prove/disprove that $L_1$ and/or $L_2$ are decidable

Question: Let $L_1$ and $L_2$ be languages over the alphabet $\Sigma$. If $L_1 \cap L_2$ is decidable, then $L_1$ is decidable or $L_2$ is decidable (or they both are). Definition of a decidable ...
1
vote
0answers
59 views

Check proof of Fermat's Little Theorem

I wrote an informal proof of Fermat's Little Theorem. Can someone check to see if the reasoning is valid, and if so how I could formalize it: Theorem: $n^{p-1} \equiv 1 \pmod p$ $ n \times 2n ...
6
votes
1answer
222 views

Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. ...
1
vote
1answer
57 views

How to prove: If $a \to -\infty $ and $b$ is bounded from below by a constant $k\in\Bbb R^{>0}$, then the $a\cdot b\to -\infty$

I must proof the following, with $a: \Bbb{N} \to \Bbb{R}$ and $b: \Bbb{N} \to \Bbb{R}$ If $a \to -\infty\ (n\to\infty)$ and $b$ is bounded from below by a constant $k\in\mathbb R^{>0}$, then the ...
0
votes
1answer
27 views

Proof-checking: $a \to +\infty \wedge \exists k \in \Bbb{R}(\forall r \in \Bbb{N}(k \leq b(r))) \Rightarrow (a+ b) \to+\infty$

let be $a: \Bbb{N} \to \Bbb{R}$, and $b: \Bbb{N} \to \Bbb{R}$, I must proof the following: "$a \longrightarrow +\infty \wedge \exists k \in \Bbb{R}(\forall r \in \Bbb{N}(k \leq b(r))) \Rightarrow ...
3
votes
2answers
107 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
0
votes
2answers
481 views

Proving the Heisenberg Group is a group

I have to prove that the Heisenberg Group, \begin{pmatrix} 1&a&b \\ 0&1&c\\ 0&0&1 \end{pmatrix} where $a,b,c\in\mathbb{R}$ is a group. I am proving a group ...
2
votes
3answers
66 views

Show $\lnot(p\land q) \equiv \lnot p \lor\lnot q$

Show $\lnot(p\land q) \equiv \lnot p \lor \lnot q$ this is my solution . Check it please
3
votes
2answers
362 views

Fallacious Induction Proof that All Integers are Equal [duplicate]

We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
11
votes
1answer
217 views

Let $f$ be a cont. on $\mathbb{R}$ and define $G(x)=\int_0^{\sin (x)}f(t) dt $. Show that $G$ is differentiable on $\Bbb{R}$ and compute $G'$.

Let $f$ be a continuous function on $\mathbb{R}$ and define $$G(x)=\int_0^{\sin (x)}f(t) dt $$ Show that $G$ is differentiable on $\mathbb{R}$ and compute $G'$. This is an exercise from ...
0
votes
0answers
28 views

Showing a subspace of a Hilbert space is also Hilbert (please check my proof)

Let $V \subset H \subset V^*$ be a Hilbert triple. Let $$W = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*)\}$$ and let $$W_T = \{ u \in L^2(0,T;V) \mid u_t \in L^2(0,T;V^*) \text{ and } u(0)=u(T)\}.$$ ...
1
vote
3answers
39 views

Proof on equivalence relations help

If $a,b\in\mathbb Z$, define a relation $\sim$ on $\mathbb Z$ by $a\sim b$ iff $ab ≥ 0$. Is $\sim$ an equivalence relation on $\mathbb Z$? Proof: Reflexive: Suppose $a\in\mathbb Z$. Then ...
0
votes
2answers
37 views

Equivalence Relation Proof Help

Determine if the following relation on $\mathbb R$ is an equivalence relation: $a\sim{b}$ iff $|a-b|≤ 1$ Proof: Reflexive: Suppose $a\in\mathbb R$. Then $|a-a| = 0$ which is real and ...
0
votes
1answer
32 views

Discrete Mathematics One-to-One Proof Help

Is α: A x B --> A defined by α(a,b) = a one-to-one? (Assume A and B are ∅) This is what I have so far. Proof: Let (x1,yn) is in A x B where n is a positive integer. Suppose α(x1,y1) = x1 and ...
2
votes
0answers
261 views

Krull dimension of quotient rings

This question is very related to this other question. I have an alternative solution to the ones proposed in the answers, and I'd like to know if it is correct. I want to find the dimension of ...
1
vote
0answers
44 views

Some properties in differential geometry of curves and surfaces

Let $\beta, \alpha$ be curves in $\mathbb{R}^3$ parametrized by arc length. Suppose $\beta$ is obtained by rotating $\alpha.$ Let $t_{\alpha}, n_{\alpha}, b_{\alpha}$ (resp. $t_{\beta}, n_{\beta}, ...
2
votes
2answers
43 views

Discrete Mathematics Proof

If m is an even integer and n is an odd integer, then m+n is odd. Proof: Suppose m is an even integer and n is an odd integer. Then $m=2k$ and $n=2j+1$ for some integers k and j. So $$m+n = 2k + ...
3
votes
1answer
288 views

Showing $\sin(x) < x$ for all $x>0$ using the mean value theorem

I want to show that $\sin(x) < x$ for all $x>0$, using the mean value theorem. Since the sine is bounded above by $1$, it's obviously true for $x > 1$. Consider $x \in ]0,1]$. Let ...
1
vote
1answer
57 views

Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
1
vote
1answer
56 views

Find sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{n(n+1)}$

I need to find sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{n(n+1)}$ I'm getting it as $2\ln2-1$