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

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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
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1answer
59 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
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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
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2answers
111 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
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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
29 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
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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
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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
352 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
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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
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0answers
99 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
55 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
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1answer
74 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
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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
52 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
96 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
51 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
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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
107 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
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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
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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
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3answers
40 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 ...
0
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1answer
24 views

Finishing the proof: given any subspace, it has a complement

The proof is as followed but I was not able to complete it, I hope someone could hit it with a magic stick. Proof: Let $V, U \subset X$ be two subspaces, where $dim(V) = k$, $dim(X) =n$, $k ...
1
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0answers
17 views

Show that no set $S$ is equinumerous with $P(S)$. [duplicate]

Looking for a proof check. I've written out the argument as it reads most easily to my own eyes. Let $S$ be a set and suppose by contradiction that $\exists f:S \to P(S)$ with f surjective. Define $B ...
0
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0answers
47 views

Show the positive, non-empty set of real numbers $F$ is countable.

It is true that for every non-empty finite $A \in F$. $$ \sum_{x \in A} \leq 1 $$ By the Archimedean Property we have $x>0, \exists n \in \mathbb{N} : \frac{1}{n} < x$. Let $S = \{x \in F: x ...
0
votes
2answers
44 views

Relatively Prime Relationship Equation Proof

I have this math question that I am stuck on. This is the question: Suppose that $a$ and $b$ are positive integers, with $d = \gcd(a, b)$. Suppose that there exists integers $r$ and $s$ so ...
2
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2answers
79 views

Proving Theorems

I've been struggling with the concept of proofs ever since I completed my introductory logic course "Axiomatic Systems". In that course it seemed to be easy. We were pretty much just using various ...
0
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0answers
34 views

Proof that the collection of all finite subsets of $\mathbb{N}$ is countable.

Just looking for some feedback on my proof. Am I missing anything? Thanks! Suppose $A$ is the collection of all finite sets in $\mathbb{N}$. Trivially we have $\emptyset = 1$. Define function f on A ...
0
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1answer
21 views

Show simple continued fraction with Euclid's Algorithm

I have this math problem, I have to show a simple continued fraction from equations. Here's the question. Use these equations: $$397 = 204(1) + 193$$$$204 = 193(1) + 11$$$$193 = 11(17) + 6$$$$11 ...
1
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1answer
16 views

Prove with Euclid's Algortihm

I have this problem and I'm not 100% how to complete it. Here's the question: Let $m$ and $k$ be positive integers with $m > 1, k > 1$. Show that $\gcd(m, mk - 1)=1$. (Hint: use ...
1
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0answers
25 views

Prove that $AB\mid CD$

I have this math question that I'm kind of confused on. This is the question: Let $A, B, C$ and $D$ be integers with $A \mid C$ and $B \mid D$ show that $$ AB \mid CD. $$ I'm not 100% sure ...
3
votes
3answers
92 views

Prove using induction

I have this math problem I'm kind of stuck on. Here's the question: Define a sequences of real number with the definitions $$\begin{align*} x_1 & = 3 \\ x_n &= \sqrt{2 x_{n-1}+1} ...
0
votes
3answers
61 views

$\int_0^c f(x)dx=0$ for each $c\in[0,1]$ then $f=0$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous and $$\int_0^c f(x)dx =0$$ then prove that $f\equiv 0$.Do not assume that $f\ge 0$. How do I do it $?$ . I have this slightly geometric ...
0
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1answer
14 views

proofs involving the triangle inequality.

$$\forall a,b \in R (|a+b|=|a|+|b| \iff ab \ge 0)$$ I'm really stuck on where to even start with this. I'm assuming it has something do to with the triangle inequality, but don't know how to apply ...
0
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2answers
44 views

Strange induction proof

I'm trying to solve an induction proof exercise but this time I can't even understand how to proceed. I must prove that for every given $n\in \mathbb{N}$ with $n\geq2$ there exist $a,a_1,a_2,...,a_n$ ...
1
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1answer
43 views

What continuity of the derivative has to do with the proof of $\int_a^b f +\int_{f(a)}^{f(b)} f^{-1} =bf(b)-af(a)$

Question: Let $f:[a,b]\rightarrow \mathbb R$ be continuously differentiable and $f'\gt 0$. Prove that $$\int_a^bf +\int_{f(a)}^{f(b)} f^{-1} = bf(b)-af(a) $$ My problem with this : $f'\gt ...
1
vote
4answers
51 views

counterexample to a proof.

Prove the following statement; $$\forall a,b \in R (\forall \epsilon > 0 (a \le b + \epsilon) \rightarrow a \le b)$$ I can't see how this is true This means that I can pick a number for all ...
3
votes
2answers
32 views

Show that all elements of one sequence are less than all elements of another sequence.

Let $\{a_n\}_1^\infty$ and $\{b_n\}_1^\infty$ be two sequences in $\mathbb{R}$ such that $\forall n \in \mathbb{N}$, it is true that $a_n \leq b_n, a_n \leq a_{n+1}, \text{and} b_{n+1} \leq b_n$. We ...
3
votes
2answers
86 views

Show that sup$AB$=(sup$A$)(sup$B$)

Where $AB$ is the product of the sets and $A,B \in \mathbb{R^+}$. Since $A,B$ are bounded above sup $A$ and sup $B$ exist. Let $\alpha = $ sup $A$ and $\beta = $ sup $B$. This implies $\forall a \in ...
0
votes
2answers
66 views

How to prove prime power factorization is square free

The question is as follows: "Show every positive integer is the product of a square (possibly 1) and a square free integer" We begin by writing a positive integer n in its refined prime power ...
1
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1answer
42 views

Strengthening an inequality

I'm reading a book and there's an example problem that goes like this: Prove that $$ \left(\frac{1}{2}\right) \left(\frac{3}{4}\right) ... \left(\frac{2n-1}{2n}\right) \le \left(\frac{1}{\sqrt ...