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

learn more… | top users | synonyms

0
votes
1answer
11 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 ...
4
votes
4answers
283 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 = ...
1
vote
3answers
36 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?
1
vote
7answers
61 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 ...
0
votes
0answers
11 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
votes
2answers
22 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 ...
2
votes
1answer
51 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
votes
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 ...
0
votes
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 = ...
2
votes
1answer
31 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 ...
1
vote
2answers
27 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 + ...
1
vote
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 ...
0
votes
1answer
16 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
votes
1answer
20 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 ...
0
votes
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 ...
0
votes
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?
0
votes
0answers
17 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
votes
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 ...
0
votes
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 ...
1
vote
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?
3
votes
3answers
35 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 $$X \setminus V = U \cap X $$ ...
1
vote
3answers
28 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 ...
3
votes
6answers
118 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?
1
vote
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
vote
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
votes
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
votes
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 ...
0
votes
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 ...
2
votes
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, ...
1
vote
2answers
29 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 ...
0
votes
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 ...
0
votes
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."?
0
votes
2answers
41 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
votes
1answer
44 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
votes
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
votes
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 + ...
1
vote
0answers
24 views

Prove $\sqrt s$ exists by the Intermediate Value Thorem?

I'm pretty sure the process is right, but I'm not sure if I've validly presented the proof. Here is what I wrote: Suppose that $s > 0$ and consider the function $f(x) = x^2$ on the interval ...
1
vote
0answers
20 views

Euclids Lemma p|abc

I'm hoping yall can let me know if this proof looks okay. I'm trying to prove "If p|abc then p|a or p|b or p|c" This is what I came up with for the proof:
1
vote
1answer
47 views

Prove that there doesn't exist prime numbers $a, b, c$ s.t. $a^2=b^2+c^3$

I first showed that if $a,b,c \neq$ 2, then they are odd and therefore are never equal. Then I consider the cases where $a=2$, $b=2$ and $c=2$. It seems to be unnecessarily long so is there a more ...
3
votes
1answer
28 views

Proving that maximizing a sum of functions of different independent variables is equivalent to maximizing each function

Let $$ \pi = f_1(x_1) + f_2(x_2) + f_3(x_3) + \dots + f_n(x_n) = \sum_{i=1}^n f_n(x_i) $$ where $f_i$ denote different functions and $x_i$ denote different independent variables Would proving that ...
0
votes
1answer
28 views

Proof for length of graph

G is a simple graph that consists of a vertex set V(G) = {v1, v2, ..., vn} and an edge set E(G) = {e1, e2, ..., em} where each edge is an ordered pair of vertices. The edge {u,v} is denoted uv. A ...
0
votes
1answer
18 views

Proof for Theorem of Upper and Lower Bounds On Zeroes of Polynomials

I'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof: Let $f(x)$ be a polynomial with real coefficients and a positive leading coefficient. ...
3
votes
1answer
40 views

Is there any way to gain some insight into a proof by simply looking at a graphic?

My school is using Pinter's "A Book of Abstract Algebra" for both semesters of Modern Algebra. For a class assignment a couple weeks ago, regarding rings, I was tasked with the following problem ...
1
vote
3answers
138 views

How to minimize $ab + bc + ca$ given $a^2 + b^2 + c^2 = 1$?

The question is to prove that $ab + bc + ca$ lies in between $-1$ and $1$, given that $a^2 + b^2 + c^2 = 1$. I could prove the maxima by the following approach. I changed the coordinates to spherical ...
-2
votes
1answer
51 views

Prove or disprove these statements. [closed]

I have this statement and I need to prove or disprove it. Any help is appreciated. (1) Is it possible for solution set of a system [A| $\vec{b}^.$] of three equations and three variables, and ...
0
votes
1answer
91 views

Proving by induction that a balanced strings of parentheses has equally many opening and closing parentheses

In computer science, a string is a finite sequence of characters. For strings $A$ and $B$, we express $AB$ as $A$ followed by $B$. A balanced string of parentheses is a string of open and closed ...
1
vote
1answer
16 views

Prove the following simple exponentiation equality.

Having trouble with the following proof. Given $b > 1, c > 0$, prove that $ \exists \; x$ s.t. $b^{x} < c$. We can't use $log$, and I have already shown that $b^{x} > c$ by using the ...
2
votes
1answer
45 views

Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...
0
votes
2answers
51 views

Prove $\log(x)^{10} < x$ (for $x > 10^{10}$)

I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$ It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.
1
vote
1answer
29 views

Understanding a proof about nested nonempty connected compact subsets

I know this question has asked to death here on MSE but I have not found a satisfactory solution. A solution found online is extremely elegant but I do not quite understand it! Given nested ...