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

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1answer
23 views

Which subsets of $l^2$ are compact?

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ State whether the following subsets ...
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1answer
10 views

Vertex deletion and chromatic number proof

Let G be a graph such that, for all vertices $a$ and $b$, $\chi(G-${$a-b$}$)=\chi(G)-2$. Prove that G is a complete graph. I started by drawing $K_5$ which has chromatic number $\chi(K_5)=5$ and ...
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1answer
37 views

Proof that every non-empty subset of a woset (X, $\leq$) has a unique minimal element.

I want to prove that every nonempty subset of a woset (X, $\leq$) has a unique minimal element. What I’m looking for: clarification and/or hints. I want to solve it on my own, but this is all the ...
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1answer
22 views

Under what conditions can we move the limit symbol through the logarithm symbol?

I was reading the derivation of the derivation of a log function. And saw this: $$\frac{d}{dx}[\log_b x]= \frac{1}{x}\lim_{v \to 0} [\log_b(1+v)^\frac{1}{v}]$$ Then, the limit notation gets moved ...
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0answers
34 views

How do I memorize mathematical proofs?

I first started wanting to know about the derivation of theorems because certain ones help you memorize the theorems better. But as I take harder math classes, it turns out better for me to use ...
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1answer
38 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...
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2answers
37 views

Algebraically transform logic expression

Algebraically transform: $\neg \forall x(P(x) \wedge Q(y) \implies \exists zR(z))$ to $\exists x\forall z(P(x) \wedge Q(y) \wedge \neg R(z))$ Justify each step with one or more ...
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4answers
74 views

Find $A$ and $B$ such that $A⊈B$ and $B⊈A$? [on hold]

I need to prove that the subset relation “$⊆$” on all subsets of $\mathbb Z$ is not a total order and I'm going to do this by finding $A$ and $B$ such that $A⊈B$ and $B⊈A$? Is there a simple solution ...
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3answers
52 views

Proof with Fibonacci Sequence

I was working on my homework assignment for one of my classes and I have come across a proof question that my classmates and I are finding difficult to answer. The problem is asking us to prove that ...
3
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1answer
39 views

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$

Show that $S_4$ is not isomorphic to $D_4\times\mathbb{Z}_3$ I have no idea how to show this. I'm studying for a test, so I am less interested in solutions and hints than I am strategy. What ...
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4answers
49 views

Details of proof by contradiction

I realize this is pretty basic but recently became unsure of how to justify proof by contradiction. Is it that case that I can show $A\Rightarrow B$ by assuming $A$ and NOT $B$, and showing this leads ...
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1answer
34 views

Logical form of a set-theoretic statement.

From Velleman's 'How to Prove it' book, there is one statement - written below - of which I don't know how to write the logical form of, and I'm wondering if somebody could write it out. The ...
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2answers
42 views

If $X$ is a compact metric space and $E_n$ is closed nonempty subset, show that $\cap_{n=1}^\infty E_n$ is nonempty.

Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is ...
1
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1answer
31 views

If $a$ is the only element of order $2$ in a group, show that $a$ is in the center of the group.

If $a$ is the only element with order $2$ in a group $G$, then $a \in Z(G)$. I'm studying for a test and I can't figure out how to prove it. What kind of methods might I try to solve this ...
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1answer
47 views

Subset relation ⊆ on all subsets of ℤ is a partial order, not a total order.

I need to prove that the subset relation “⊆” on all subsets of ℤ is a partial order but not a total order. I'm not experienced in these kind of proofs and was hoping to see an example of an easier one ...
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4answers
295 views

Definition of Inverse in Linear and Abstract Algebra

In a linear algebra text, the following is the definition of the inverse of a matrix An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = ...
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3answers
39 views

How to prove that $x^2 + 3y^2 = 1$ is contained inside of the unit ball?

What is the best way to show that $S = \{(x,y) | x^2 + 3y^2 = 1\}$ is contained in the unit ball without graphing the set?
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7answers
72 views

Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
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0answers
13 views

$∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$

I am trying to prove the following statement: $∀ Z^+$ can be written as $c_r * 3^r + c_{r-1} * 3^{r-1} + …. + c_2 * 3^2 + c_1 * 3 + c_0$ where $c_r =$ 1 or 2, and $c_i$ = 0, 1, or 2 for all integers ...
0
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2answers
25 views

If $G$ is isomorphic to $H$, show ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$

For every $\alpha\in{\rm Aut}(G)$, I've defined $A:H\rightarrow H$ by $$A(h)=\phi(\alpha(\phi^{-1}(h)))$$ where $\phi$ is an isomorphism from G onto H, and I've shown that $A\in {\rm Aut}(H)$. What I ...
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1answer
53 views

Homeomorphism between $\mathbb{R}$ and $\mathbb{Q}$ - why does cardinality matter?

When I look up why $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, almost all the answers just say something along the line of "Because, Cardinality" and then ends there. Can someone provides ...
3
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1answer
44 views

What Sort of Discovery Warrants Writing a Paper

I am a high school student who is deeply passionate about mathematics and I have written many different mathematical proofs. I was wondering what sort of discovery warrants writing a mathematical ...
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1answer
24 views

Why is the following reverse triangle inequality true for given series?

I wish to show that for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$ then claim : $|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty a_k| - |\sum\limits_{k = ...
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1answer
32 views

Prove that $|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$

Prove for any $a, b \in [-1, 1]$ that $$|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$$ I'm sure there is a solution using the Cauchy-Swartz inequality. Thus i tried to ...
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2answers
28 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
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2answers
25 views

Universal Instantiation and Proof of: if $x$ and $y$ are odd integers, then $xy$ is odd.

I have a question about the first part of the proof for the statement "If $x$ and $y$ are odd integers, then $xy$ is odd" if we are using a direct proof. Now i've gotten the proof roughly correct but ...
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1answer
17 views

Direct proofs involving disjunctions

I've just started a logic and proof class, and I'm confused about what we learned. Given a proof of the form $$(P \lor Q) \rightarrow R$$ why is it true that you only have to show $$P \rightarrow R$$ ...
0
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1answer
21 views

How to prove that the steeple function is not uniformly convergent?

In class we encountered this function $$f_n(x)=\begin{cases} n^2x, & 0 \leq x \leq 1/n\\ 2n - n^2x, & 1/n \leq x \leq 2/n\\ 0, & 2/n \leq x \leq1 \end{cases}$$ The prof said ...
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2answers
14 views

Each regional of a maximal planar graph is a triangle

Show that every region of a maximal planar graph is a triangle. A planar graph G is called maximal planar if the addition of any edge to G creates a nonplanar graph. Will this proof use the ...
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0answers
35 views

Geometry perpendicular proof

How would I prove that there is a line perpendicular to any given line through a given point not on the line?
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0answers
21 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
0
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1answer
33 views

Find the limit of $P_{\theta_n}\Big(\sqrt{n}(T_n-\mu(\theta_n))<z_\alpha \sigma(0)-\sqrt{n}(\mu(\theta_n)-\mu(0))\Big)$

Assumptions: Consider a sample of i.i.d random variables $X_i$ $i=1,...,n$, where each $X_i$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_i:\Omega\rightarrow ...
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0answers
10 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
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2answers
42 views

If $(d,a)=1$ and $d|ab$ then $d|b$ .

Okay, checking to see if i'm on the right track. I essentially did the same prove for Euclid's lemma but exchanged the $d$ for the $p$. Is that the right idea? Or am I missing something?
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3answers
36 views

How to prove these two sets are identical?

This is more a question of the methadology one should use to solve these type of questions: Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X $$ ...
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3answers
30 views

{students 1 and 2 are in different groups} vs {students 1, 2, 3, and 4 are in different groups}

Source: Example 1.11, p 26, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Hereafter abbreviate graduate students to GS and undergraduate students to UG. Example 1.11. A ...
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6answers
122 views

If $p$ is prime then $2p+1$ cannot be square

How can I prove that $2p+1$ cannot be a square number if $p$ is prime? Is a contradiction proof enough where I assume true then show it as false eventually?
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1answer
25 views

eigenvalue diagonal

Suppose that $B$ is an element of $M_n(F)$ and that for each $j = 1,\dots,n$ $e_j$ is an eigenvector of $B$. Prove that $B$ is a diagonal matrix. I know that the basic idea involves showing that ...
1
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1answer
64 views

Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$

Let $V$ be a vector space over a field $k$ and let $U$, $W$ be finite-dimensional subspaces of $V$. Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$ I'm given that to begin this ...
2
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1answer
15 views

Applying the Dimensional Formula to Prove a Corollary

I am to prove the following corollary. Let $V_1$ and $V_2$ be subspaces of a $n$-dimensional vector space $V$. If the sum of the dimensions of $V_1$ and $V_2$ is greater than $n$, then $V_1$ and ...
0
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1answer
44 views

non-zero divisors in a ring

I am asked to show the following: $ab$ is a non-zero divisor of $R$ if and only if $a$ and $b$ are both non-zero divisors of $R$. $\Rightarrow)$ Suppose $ab$ is a non-zero divisor of R. Then ...
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2answers
40 views

Prove $(a_1 + a_2 + a_3)^2 = a_1^2 + a_2^2 + a_3^2 + 2(a_1a_2+ a_2a_3 + a_1a_3)$

I wish to find a proof for this equality: $(a_1 + a_2 + a_3)^2 = a_1^2 + a_2^2 + a_3^2 + 2(a_1a_2+ a_2a_3 + a_1a_3)$ But then I realized there exists a more general version: $(\sum\limits_{k=1}^n ...
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0answers
29 views

Functions disagreeing on a set of measure zero - Proof verification

I was asked to prove that if $f\in\mathscr{R}$ on some compact rectangle $Q\subset\mathbf{R}^{n}$ and if the set $D=\lbrace \mathbf{x}:\mathbf{x}\in Q,\,f(\mathbf{x})\neq 0\rbrace$ has measure zero, ...
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2answers
41 views

Deductive Proof - Justify each step with law or inference rule

My Professor gave me the following: a) If $P \to Q, \neg R \to \neg Q$, and $P$ then prove $R$. b) If $P \to (Q\wedge R)$ and $\neg R\wedge Q$ then prove $\neg P$. I understand how to do ...
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1answer
35 views

Proofs that include long terms

While writing a proof that included expansion of $(\sqrt{n} + 1)^8$ in which I know the first term has to be $n^4$, is it acceptable to write something like $(n^4 +...... + 1)$ if the $n^4$ is the ...
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0answers
48 views

Is there any difference between “for any” and “for all”?

When we prove something, we use mathematical symbol ∀ to stand for "for all." Does it make any difference if we use same symbol for "for any."?
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2answers
42 views

Let A ={x ∈ Z | x =18a − 2 for some integer a } and B ={y ∈ Z | y = 18b + 16 for some integer b} Prove A ⊆ B

I'm fairly certain that this proposition is true, but I have no idea how to approach a proof for it. I'm not looking for someone to do the work for me, I'm just trying to find out what type of proof ...
0
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1answer
46 views

Proof check/ suggestion: The suspension of $S^n$

In one of my excercise sheets there was a remark saying that $$SX \approx S^{n+1}$$ where $SX$ denotes the suspension of $X=S^n$. So I tried to prove this on my own and would like to discuss my ...
0
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1answer
24 views

Proof with area segments in a triangle

I have to show that $A M_CS$ and $M_CBS$ have the same area $X$ and that concerning areas $X=Y=Z$ is true. I'm really stuck here, I would appreciate any help or tip...! How can I start here?
0
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1answer
36 views

Zero divisors within the ring of dual numbers

Let $\mathbb{R}(\epsilon) = \{ a + b \epsilon : a,b \in \mathbb{R} \}, $ where $(a + b\epsilon) + (c + d\epsilon) = (a+c) + (b+d)\epsilon$ and $(a+ b\epsilon) \cdot (c + d\epsilon) = ac + (ad + ...