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

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0
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2answers
61 views

Proof that $|d(x,y) + d(y,z)| \leq d(x,z)$

Here there is my proof (quite short and easy) of a rather straightforward result. Still, I would like to know: if it is sound, because absolute value always creates me some problem, and if there ...
0
votes
2answers
34 views

Least upper bound of $A*B$ [duplicate]

Let $A$ and $B$ be sets of positive real numbers. Let $A*B = [ab$ | $a \in A$ and $b \in B]$. Then $lub(A*B)=lub(A)*lub(B)$ I think this is true, but I'm not sure how to prove it, and have been ...
0
votes
0answers
17 views

Principle Branch of Derivative for Arcsecant Function

In class, we defined the inverse secant function as $$y=arcsec(x) \iff secy=x, y\in\ [0, \frac{\pi}{2}) \cup [\pi,\frac{3\pi}{2})$$ With this choice of principal branch the derivative of ...
0
votes
0answers
20 views

A proof regarding bump functions and integration

How do I prove this statement: If $I: C_0^{\infty}(\mathbb{R}^n)\rightarrow\mathbb{R}$ is linear and positive (that is for all $f\geq 0$ holds: $I(f)\geq 0$) and there exists a constant $C_1$ such ...
1
vote
0answers
32 views

Proving a bump function

Let $\phi$ be a bump function on $\mathbb{R}^{n+m}$. How do I prove that the function $u(x)=\int_{\mathbb{R}^m}\phi(x,y)$d$y$ is a bump function on $\mathbb{R}^n$?
1
vote
5answers
148 views

Proving that $64$ divides $3^{2n+2}+56n+55$ by induction

Let $n ≥ 0$ be an integer. Prove by induction: 64 divides $3^{2n+2} + 56n + 55$ general expression: $3^{2n+2} + 56n + 55 = 64m$ 1st I substitute $P(0)$ and it gives me true: $9+55 = 64$ (if m = 1 ...
0
votes
1answer
34 views

How to prove a theorem in logic?

Let $SF=(V,Wffs,A,R)$ be a formal system defined by: $V=\{ 1, +,=\}$ $Wffs=\{1^m+1^n= 1^p \ with \ n,m,p>=0\}$ $A=1+1=1^2$ ($A$ stands for axiom) $R=\{r_1,r_2\}$ $ \ \ \ ...
1
vote
1answer
39 views

Show that $f(x)$ is continuous

$f(x)=\begin{cases}2x\,\,\,\text{if $x\geq 0$}\\-3x\,\,\,\text{if $x<0$}\end{cases}$ Show that $f(x)$ is continuous. To show $f(x)$ is continuous, since I haven't covered $\epsilon-\delta$ ...
1
vote
3answers
81 views

Prove $\bigcap \{A,B,C\} = (A \cap B) \cap C$

This is a follow-up to a previous question where I asked how to prove $\bigcup \{A,B,C\} = (A \cup B) \cup C$. The proof of $\bigcap \{A,B,C\}=(A \cap B)\cap C$ is mostly analogous: \begin{align*} x ...
0
votes
2answers
49 views

If $0 \in P$ and $\sup\{x \in [0,1] : [0,x] \subseteq P\} = 1$, then $[0,1] \subseteq P$? (What if adding 3rd assumption?)

Let $P \subseteq \mathbb{R}$. If $0 \in P$ and $\sup\{x \in [0,1] : [0,x] \subseteq P\} = 1$, then $[0,1] \subseteq P$. Is the above claim true or false? How to prove or disprove it? Edit1: ...
7
votes
1answer
47 views

Is this proof that *on average it takes $e$ straws to break the camel's back* sufficient and clear?

Problem: There is a camel with a tolerance of weight 1. We stack straws on his back one by one, and the weight of each straw is a number randomly sampled from a uniform distribution between 0 and 1. ...
1
vote
1answer
53 views

$x^3+y^3<1<x+y \implies (x,y)\in{]0,1[}$

We have : $H$ is the set $\{(x,y)\in \mathbb{R}^2 | x^3+y^3<1<x+y\}$. How can one show that the set $H$ is included in ${]0,1[}\times{]0,1[}$ ? We have : $(x+y)(x^2-xy+y^2)<1<x+y$ ...
0
votes
2answers
37 views

Show that $y^n+1/y^n \in \mathbb{N}$

Let $y$ be the solution of the equation $x+\frac{1}{x}=3$ such that $y>1$. Show that $y^n+\frac{1}{y^n} \in \mathbb{N}$ We have $x+\frac{1}{x}=3$ if and only if $ x^2-3x+1=0$. Which has two ...
1
vote
2answers
45 views

How to prove that $4x \le x^2 + 8$ for all $x$

I need to find a way to prove the above statement, and feel that doing it by individual cases is not the best method.
0
votes
3answers
46 views

Linear Algebra proof, show if a matrix is invertible

Let $n$ be a natural number $\geq 2$ and $A$ a matrix $\epsilon M_{n}(K)$. We suppose the matrices $A$ and $I_{n}+A$ are invertible. Show that $I_{n}+A^{-1}$ is invertible and also $A(I_{n}+A)^{-1}$ ...
1
vote
1answer
34 views

Proof regarding the union of two subgroups (abstract algebra)

Given a group $G$ and two subgroups $H_1\leq G$ and $H_2\leq G$. Also: $H_1\cup H_2= G$. I have to prove, that either $H_1=G$ or $H_2=G$. So, if the group $G$ is the union of the two subgroups ...
-1
votes
2answers
51 views

Summation proof- struggling to see a way to prove

I have found that the summation attached gives a general value of 1/(n+1) for the first few values of n=0,1,2,3.... I would like to prove that this is true for all n and I assumed that the best way ...
0
votes
1answer
28 views

How to prove that sentence using constructive proof?

Ok, I admit it's homework, I am supposed to prove some sentences using natural logic, I can do some, but I'm stuck on this one: ¬E→(E→(E→F)) I can prove it using contradiction, but I'm supposed to ...
0
votes
1answer
24 views

Find languages L1 and L2, neither of which contains the other, such that (L1* ∪ L2*) = (L1 ∪ L2)*. [closed]

I'm trying solve this question in several ways, but only textbook has not helped me alot.
1
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4answers
60 views

Prove rational nums

For all real number x : R(x) -> there exist two integers k, l such that x = k/l. (i.e. x is a rational number) Prove/Disprove: For all real number x : R(x) -> R(x+1) My answer: Let x be a real ...
-1
votes
1answer
55 views

Associative Property of Multiplication - Cardinal Numbers Proof

For Cardinal numbers A, B, C prove that ( AB ) C = A ( BC ) I've read that A bijection between A × (B×C) and (A×B) × C can be given by (x,(y,z))↦((x,y),z) from ...
0
votes
2answers
26 views

Prove the solutions to $ax+by=c$

I have this math question, that I'm kind of stuck on. Consider the equation $a x + b y = c$, for some non-zero integers $a, b$ and $c$. Suppose that $x = x_1, y = y_1$ is an integer solution to ...
0
votes
2answers
110 views

If $A\subseteq\mathbb{R}$ such that $0$ is a limit point of $A$, is $\{ka : k\in\mathbb{Z}, a\in{A}\}$ dense in $\mathbb{R}$?

Let $A\subseteq\mathbb{R}$ such that $0$ is a limit point of $A$. Is the set $ZA:=\{ka : k\in\mathbb{Z}, a\in{A}\}$ necessarily dense in $\mathbb{R}$? P.S. Please on my current stage of learning I ...
0
votes
3answers
28 views

Let A and B be matrices of Mn(K). Show that if AB is invertible the matrices A and B are invertible.

Let $A$ and $B$ be matrices of $M_{n}(K)$. Show that if $AB$ is invertible the matrices $A$ and $B$ are invertible. So i know how to find the inverse of a matrice, i know how to do the product of two ...
6
votes
1answer
68 views

Prove that P(A) ∪ P(B) ⊆ P(A ∪ B).

I have a presentation on this Monday. I thought it was pretty straight forward but my professor wrote "You need to show why x is in P(AUB), not just state that it is." I thought that I had. Here's ...
2
votes
2answers
44 views

Matrix proof, linear algebra

Let $$R(\theta)=\begin{bmatrix} \cos\theta &-\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$ Also, $a$ and $b$ are real numbers. We suppose that $b\neq 0$ and we consider the ...
2
votes
5answers
28 views

Prove divisibility: if $a\mid (b-d)$ and $a\mid (c-e)$, then $a\mid (bc-de)$

I have this math question. It states: Show that for any $a , b ,c, d, e \in \mathbb{Z^+}$, if $a\mid (b-d)$ and $a\mid (c-e)$, then $a\mid (bc-de)$. I'm not 100% sure as to how to start this ...
0
votes
0answers
18 views

Prove Bezout Eqaution Problem

I have this math problem, that I'm kind of confused on. Consider the equation $a x + b y = c$, for some non-zero integers $a, b$ and $c$. Suppose that $x = x_1, y = y_1$ is an integer solution ...
0
votes
1answer
10 views

Prove gcd and common divisor

I have this math problem. Let $a, b, m$ be any positive integers with $\gcd(a,m)=d$ and $\gcd(b,m)=1$. i) Show that if $k$ is a common divisor to $ab$ and $m$, then $k$ divides $d$. ...
0
votes
4answers
351 views

Which matrices property should I use to prove that?

For any real number $\Theta$, we say $$R(\Theta)=\begin{bmatrix} \cos\Theta &-\sin\Theta \\ \sin\Theta & \cos\Theta \end{bmatrix}$$ Show that $(R(\Theta))^n= R(n\Theta)$ for any ...
1
vote
2answers
47 views

Show by induction: $(1+\frac{1}{n})^{n}<n$

Show by induction that for all natural numbers n>3 $(1+\frac{1}{n})^n<n$ Let $(1+\frac{1}{n})^n<n$ be true ! We show that $(1+\frac{1}{n+1})^{n+1})<n+1$ ...
1
vote
1answer
40 views

Show that if $f:A\to B$ is a bijection then $f^{-1}$ is a surjection

The proof I have for this is as follows: Let $a\in A$ then $f(a)=b\in B\Leftrightarrow f^{-1}(b)=a$ and so for $b=f(a)\in B$ and $a=f^{-1}(b)$. The math and everything makes sense to me but I don't ...
2
votes
3answers
45 views

Proof by Induction (Inequality)

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
0
votes
0answers
94 views

Proof of the Crossbar theorem

A teacher asked me to prove the well known Crossbar theorem. I tried it in the following way:- Given: If $D$ is in the interior of $\triangle ABC$, then prove that $\overrightarrow{AD}$ intersects ...
0
votes
1answer
29 views

Show that if $S \subseteq \mathbb{R}^n$ the int(S) is open, and that, if $U$ is any open subset of $S$, then $U \subseteq$ int(S).

Looking for feedback on my proof: Let $A$ be all the open balls $\left\{ x|\exists r > 0 (B(x;r) \subseteq S \right\}$ in $S$. By definition int$(S) = A$ and since for all $x \in A$ there is an ...
1
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0answers
52 views

Show that the sequence {(2n)/(n+2)} converges to 2 using the \epsilon definition

And I based my work on the following example:
0
votes
1answer
68 views

Proof completion problem: I can only use primitive rules of inference, and I have contradictory premises.

Standard proof completion: ~(p&q) A ~(~p&q) A ~(p&~q) A ~(~p&~q) A SHOW r Contradicting r and then showing a contradiction seems like the obvious plan of attack, but after that I'm ...
0
votes
1answer
39 views

Prove that a group of size $\ge18$ people can be assembled from groups of 4 and 7

How can I prove that a group of size $\ge18$ can be assembled from groups of $4$ and $7$ using the well ordering principle? Well-ordering principle: Every nonempty subset $T$ of $N$ has a least ...
-3
votes
1answer
51 views

Prove that if H ∪ K is a subgroup of G… [duplicate]

Suppose G is a group, with subgroups H and K. Prove that if H ∪ K is a subgroup of G implies that H ⊆ K or K ⊆ H. I'm not really sure how to start this, I can prove that H ∩ K is a subgroup but I ...
1
vote
2answers
95 views

Prove $\bigcup \{A,B,C\} = (A \cup B) \cup C$

Note: The analogue of this question for intersections is answered here: Prove $\bigcap \{A,B,C\} = (A \cap B) \cap C$ This question asks how to prove $\bigcup \{A,B,C\} = (A \cup B) \cup C$. ...
0
votes
1answer
27 views

Verification of a proof regarding the connected sum of two surfaces

I am trying to solve the following exercise: Let $X_1, X_2$ be two surfaces. Lets consider charts $\varphi_j: U_j \to \mathbb{R}^2$ with $U_j \subset X_j$, $j= 1, 2$ and let $B_j = ...
0
votes
4answers
49 views

proof of limit involving factorials and exponents

$ \cdot \lim \limits_{n \to \infty}\frac{10^n}{n!} $ I know intuitively that this is zero but I'm not sure how to prove this. Can I use an inequality? Maybe $\frac{10^n}{n!} \le \frac{1}{n!}$ when ...
3
votes
2answers
40 views

Proving the nested interval theorem

Theorem: Let $\{I_n\}_{n \in \mathbb N}$ be a collection of closed intervals with the following properties: $I_n$ is closed $\forall \,n$, say $I_n = [a_n,b_n]$; $I_{n+1} \subseteq ...
1
vote
1answer
30 views

$|f(x)-f(1)|<k|x-1|$

Given the function : $f(x)=x^2+x|x-1|-1$ such that $x$ is a real number . Show that there is a $k\in \mathbb{R}$ for all $x\in \mathbb{R}$ such that: $|x-1|<1 \implies |f(x)-f(1)|<k|x-1|$ I ...
0
votes
2answers
24 views

Symmetric relation proof

Prove that the following relation is symmetric: For all $x,y\in\Bbb N$, $xRy$ iff $x+y$ is even. My attempt: Assume $x,y$ are in $\Bbb N$, and $x+y$ is even. Since $x+y$ is even, then $x+y=2a$ for ...
1
vote
1answer
21 views

Induction T/F questions. How to know what the counterexample is.

Determine whether the statement is true of false. If true, provide a proof. If false provide a counterexample. for $n \in N, 2n-8 < n^2-8n+17$ I started off like a typical induction proof. ...
2
votes
2answers
106 views

Interesting areas of study in point-set topology

I'm undertaking a little self-study in point-set topology, because my undergraduate course does not have a module in Topology. I have a copy of Topology by James Munkres, but do not have the time to ...
1
vote
0answers
22 views

How to show that $\int_{\delta D} x\ dx $ is area of $D$

Prove that $\int_{\delta D}x\ dy$ is area of the $D$ and $\int_{\delta D}y\ dx$ is munis the area of $D.$ Now using Green's Theorem I can prove that $$\int_{\delta D}x\ ...
1
vote
1answer
78 views

Prove that definitions of the limit superior are equivalent

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence. And let $L^+$ be an extended real number (i.e. $L^+\in\mathbb{R}^*$). Then TFAE: (1) $L^+$ = ...
1
vote
3answers
39 views

Multiple logical quantities for English statement?

Express each of the following statements as a conditional statement in "if-then" form or as a universally quantified statement. Also write the negation (without phrases like "it is false that") g) I ...