For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.
2
votes
1answer
32 views
simple proof for logical formula
I am stuck in this proof, I am given:
$$A\setminus(B\setminus(C\setminus D)) = (A\cup C)\setminus (B\cup D)$$.
I did this, but cannot come to solution where i can say, this is true or not.
...
4
votes
2answers
29 views
$S=\{(X,Y) \in \mathcal{P}(A) \times \mathcal{P}(A) \mid \forall x \in X \exists y \in Y(xRy)\}.$ If R is symmetric, must S be symmetric?
I'm working on an exercise from How To Prove It by Velleman, and I'm having a hard time.
Suppose $R$ is a relation on $A$ and define a relation S on $\mathcal{P}(A)$ as follows: $$S=\{(X,Y) \in ...
2
votes
1answer
35 views
Suppose $R$ and $S$ are transitive relations on $A$. Prove that if $S \circ R \subseteq R \circ S$ then $R \circ S$ is transitive.
Suppose $R$ and $S$ are transitive relations on $A$. Prove that if $S \circ R \subseteq R \circ S$ then $R \circ S$ is transitive.
First, I'm wondering if my proof is correct? Second, I'm really ...
2
votes
1answer
71 views
Hard-wiring a proof method in my head
There's s a kind of proof regularly used in linear algebra ( proving facts about Transformations, direct sums, basis, ... ) that i have definitely agreed with but still couldn't connect my intuitive ...
0
votes
1answer
49 views
Proving recurrence relations
So, I initially proved the theorem that if $a != b^d$ and $n$ is a power of $b$, then $f(n) =
C_1n^d + C_2n^{log_b a}$, where $C_1 = b^dc/(b^d − a)$ and $C_2 =
f(1) + b^dc/(a − b^d )$.
This is seen ...
5
votes
2answers
70 views
How to exactly write down a proof formally (or how to bring the things I know together)?
I've always had trouble with this: I know everything I need to know but I can't connect everything I know in the way it's being wanted from me.
This is what I have to do:
Prove for $f : M → N$ the ...
1
vote
0answers
49 views
Mathematical induction
So the question was basically "
Suppose that there are n teams in a rugby league competition. Every team A
plays every other team B twice, once at the home ground for team A, and the other time
at the ...
-4
votes
1answer
54 views
Transformation Existence Proof: A Call for Critique [duplicate]
QUESTION
Prove that there exists a $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
ATTEMPTED ANSWER
Let $V$ and $W$ be finite-dimensional vector spaces over $F$. Let ...
2
votes
1answer
32 views
proof of $p'(x)^2 \geq p(x)p''(x) \text { for all x } \in \Bbb{R}$
$p(x)$ is non-constant polynomial with only real roots. If $x = a_i$ is a root of $p(x)$, we are done. Assume then $x$ is not a root. Product of differentiation:
$$p\prime(x) = ...
3
votes
1answer
99 views
Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$
Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$.
I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
0
votes
1answer
34 views
Using The Pigeon-Hole Principle
Let n be a positive integer. Show that in any set of n consecutive integers there is exactly one divisible by n.
Here is the solution:
Let $a,~a+1,...,a+n-1$ be the integers in the sequence. ...
2
votes
1answer
39 views
Confused as to how to prove the basis of dft is orthonormal
I have been stuck for hours trying to prove that the basis of discrete fourier transform is orthonormal can anyone point me in the direction of how to do so
3
votes
0answers
64 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
0
votes
0answers
54 views
Show that $P = Q^2$
Suppose $P$ is a positive semi-definite $n\times n$ matrix. Show that there exists a unique positive semi-definite matrix $Q$ such that $P = Q^2$.
In class we've been going over singular value ...
0
votes
1answer
38 views
Prove the formula for the inverse of a matrix
Assuming that the matrix $A = ||a_{ij}||_{1 \leq i,j \leq n}$ is invertible, write down the explicit formula for the inverse matrix $A^{-1} = ||b_{ij}||_{1 \leq i,j \leq n}$. Prove that this ...
0
votes
2answers
47 views
Question on proof of when you add two rows on a matrix, why does $\det(B) = \det(A)$
Applying to a square matrix $A$ the row operation $R_i + \alpha R_j$ (that is, adding to Row $i$ a multiple of Row $j$), we obtain a new matrix $B$. Prove that $\det(B) = \det(A)$.
This is the ...
4
votes
3answers
81 views
Questions on the proof of “If you switch two rows on a matrix, then $\det(B) = - \det(A)$”
Applying to a square matrix $A$ the row operation $R_i \leftrightarrow R_j, i \neq j$ (that is, swapping the $i$-th and $j$-th rows), we obtain a new matrix $B$. Prove that $\det(B) = - \det(A)$.
...
2
votes
1answer
65 views
Short proof of Hall's theorem
Studying the proof of Hall's theorem in my book I started to wonder if there is a shorter way to prove it. Following is an attempt that I think works but (being short) makes me wonder if I made a ...
2
votes
1answer
35 views
Product of moscow spaces
Let $\{X_a : a\in A\}$ be a family of topological spaces such that $X_K=\prod\limits_{a\in K}X_a$ is a Moscow space of countable $o$-tightness, for every finite subset $K$ of $A$.
Then the ...
3
votes
2answers
42 views
Solving two simultaneous recurrence relations
If we have the two recurrence relations $$a_n = 3a_{n-1} + 2b_{n-1}$$ $$b_n = a_{n-1} + 2b_{n-1}$$ with $a_0 = 1$ and $b_0 = 2$.
My solution is that we first add two equations and assume that $f_n = ...
2
votes
2answers
53 views
eccentricity in vertex transitive graphs
I am trying to prove the following..
If $G$ is a veretx transitive graph, then how can we prove that eccentricity of every vertex is same? Getting no idea from where to start? How to prove the same ...
0
votes
1answer
38 views
subproof from theorem of Polya
Suppose we have polynomial:
$$f(z) = z^n + b_{n-1}z^{n-1} + \cdots + b_0$$
It is a complex polynomial of degree $\geq 1$ with leading coefficient $1$. Associate with $f(z)$ set:
$$C := \{z ...
1
vote
1answer
30 views
Why no cut-vertices or cut edges in a graph where eccentricity is same for all vertices
I need help to prove the following statement.
There are no cut-vertices or cut-edges(bridges) in a graph where eccentricity is same for all vertices. I am getting that if the graph contains a ...
1
vote
1answer
57 views
How do I check the triangle inequality using subsets?
How do I establish the triangle inequality $|x+y| \geq |x|+|y|$ by considering all real numbers as the union of six subsets and checking the inequality on each of those subsets?
0
votes
0answers
33 views
Proof of an intuition for equivalent definitions? (2)
I apologize for the double-post, but I asked the absolute wrong question and received a correct answer for that incorrect question. My apologies.
I have a very strong intuition about the ...
0
votes
1answer
20 views
Proof of an intuition for equivalent definitions?
I have a very strong intuition about the equivalency of the following definitions for Cauchy sequences:
$$\forall \varepsilon > 0, \exists M \in \mathbb{N} : k, l \geq M \implies d(x_k, x_l) < ...
2
votes
1answer
68 views
If $A$ is an invertible skew-symmetric matrix, then prove $A^{-1}$ is also skew symmetric
Let $A$ be an invertible skew-symmetric $(2n \times 2n)$-matrix. Prove that $A^{-1}$ is also skew-symmetric. (You may assume that $(AB)^T = B^TA^T$).
I did this with a $2 \times 2$ matrix and got ...
32
votes
2answers
979 views
On a 500 page proof
On wikipedia there is a claim that the Abel–Ruffini theorem was nearly proved by Paolo Ruffini, and that his proof spanned over $500$ pages, is this really true? I don't really know any abstract ...
0
votes
1answer
38 views
Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$
I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with
$$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$
First thing I want to ...
3
votes
2answers
72 views
Question of style: equiv vs. equals
Sometime I have trouble discerning whether an $=$ or an $\equiv$ is most appropriate. I believe that $\equiv$ is typically used when a new definition is being introduced, rather than a statement ...
1
vote
1answer
43 views
Prove that a multilinear function $f$ is skew-symmetric if and only if $f = \mathrm{Alt}(f)$
Prove that a multilinear function $f$ is skew-symmetric if and only if $f = \mathrm{Alt}(f)$.
I said the first thing is to prove that $\mathrm{Alt}(f)$ is skew-symmetric. In other words, we want ...
2
votes
2answers
63 views
Prove the third isomorphism theorem
I'm trying to prove the third Isomorphism theorem as stated below
Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$
I look up for some ...
5
votes
3answers
148 views
Prove the following using induction on n (matrices)
Prove the following using induction on n:
$$\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}^{n} = \begin{pmatrix} n+1 & n \\ -n & -n+1 \end{pmatrix}$$
I know that multiplication of ...
23
votes
2answers
451 views
English words in written mathematics
I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and ...
1
vote
2answers
40 views
Prove that a greedy algorithm selects the maximum number of programs
This is a homework problem. Intuitively, I know it to be true, because the largest group of programs (say, $j$ programs) must be composed of the smallest $j$ programs. But how to go about formally ...
0
votes
3answers
63 views
Prove A perpendicular to B if |a-b| = |a+b|
I'm absolutely exhausted working on this problem. I think I am very close, but I don't know what to do from here. I've tried a lot of different algebraic approaches.
So the problem says to prove A ...
1
vote
1answer
134 views
$T:P_n(F) \rightarrow F$ PROOF OUTLINE
I'd like some heavy critique if you don't mind. See here for more details.
Let $S=\{f \in P_n(F) : f(1)=0\}$. Clearly, the polynomial $f(x)=0 \in S$ because $f(c)=0$ for any choice of $c\in F$. To ...
0
votes
1answer
31 views
Correctness of Analysis argument with Cauchy sequences
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$
Proof $\rightarrow$
Suppose $(x_n) \to a$ and $(y_n) ...
1
vote
3answers
44 views
Can someone check the solution to this recurrence relation?
Here's the recurrence relation: $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$
Here's the solution:Write:
$$
a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^n + n + 3 \quad a_0 = 1, a_1 = ...
0
votes
2answers
42 views
Finding this solution to a recurrence relation
So, I know that the recurrence relation $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ has the solution of $a_n = -4(2^n) - n^2 / 4 - 5n / 2 + 1/8 + (39/8)(3^n)$. I just ...
0
votes
1answer
22 views
How to show all solutions for a particular recurrence solution
I've found that the recurrence relation $a_n = 4_{an−1} − 4a_{n−2} + (n + 1)2^n$ has the solution of $an = 2^n(p_0 + p_1n + n^2 + n^3/6)$. I'm just trying to understand the steps necessary to solve ...
2
votes
0answers
233 views
The Dinitz Problem - proof
This theorem is the one that the proof is for
Consider $n^2$ cells arranged in an $(n × n)$-square, and let $(i, j)$
de- note the cell in row $i$ and column $j$. Suppose that for every
cell ...
2
votes
2answers
53 views
Finding the solution to this specific recurrence relation
What would be the solution to $a_n = 7a_{n−2} + 6a_{n−3}$ with $a_0 = 9$,
$a_1 = 10$, and $a_2 = 32$
I can find it for a specific value of (n), but not for just a general solution. Thanks!
1
vote
1answer
53 views
Proof Technique and Factorials
I need to prove that $\;n!+m$ is divisible by $m$ for all integers $n \ge 2$ and $1 \le m \le n$.
1
vote
0answers
35 views
Proving a specific recurrence relation theorem
I'm trying to come up with a proof for this theorem:
Let $c_1$ and $c_2$ be real numbers with $c_2 != 0$. Suppose that $r^2 - c_1 r - c_2 = 0$ has only one root, $r_0$. A sequence $\{a_n\}$ is a ...
1
vote
2answers
44 views
Finding a solution to a recurrence relation
Find the solution to $$a_n = 5a_{n−2} − 4a_{n−4}$$ with $$a_0 = 3$$
$$a_1 = 2$$ $$a_2 = 6$$ $$a_3 = 8$$
My answer: Observe that the degree of recurrence is 4. Hence, the characteristic equation is: ...
1
vote
0answers
73 views
Is this theorem proof correct?
I'm trying to prove this theorem:
Let $c_1$ and $c_2$ be real numbers with $c_2 \ne 0$. Suppose that $r^2 − c_1r − c_2 = 0$ has only one root $r_0$. A sequence $\{a_n\}$ is a solution of the ...
5
votes
2answers
71 views
Product of a family of spaces of countable tightness
I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem?
Finite family of compact spaces of countable ...
1
vote
1answer
38 views
Inverse implies surjection and follow-your-nose proofs
(I'm posting this question with my own answer, to show a nice calculational proof for one of the examples in Luke Palmer's blog post Follow Your Nose Proofs.)
In what follows, $A$ and $B$ are sets, ...
0
votes
0answers
18 views
Suggestions or comments for improving this proof
For a class paper I have written the following proof:
Given an array $a$ of size $n$ and $n = 2^{\lceil \log_2(l) \rceil}$, i.e. the next greater power of 2 of $l$. It follows that:
$$2^{\lceil ...
