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
2answers
58 views
Proving if $a_1\equiv b_1\pmod{n}$ and $a_2\equiv b_2\pmod{n}$ then we have $a_1+a_2\equiv b_1+b_2\pmod{n}$.
How can I prove if $a_1\equiv b_1\pmod{n}$ and $a_2\equiv b_2\pmod{n}$ then we have $a_1+a_2\equiv b_1+b_2\pmod{n}$? I have tried several ideas I've found online but don't really understand them. Is ...
2
votes
2answers
43 views
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$ [duplicate]
Prove that if $a, b$ are both even then $\gcd(a,b) = 2\cdot\gcd(a/2,b/2)$.
Little confused here. I have tried the following but it's basically just repeating the proof unfortunately:
$a = 2 ...
1
vote
2answers
60 views
how to prove: $A=B$ iff $A\bigtriangleup B \subseteq C$
I am given this: $A=B$ iff $A\bigtriangleup B \subseteq C$. And $A\bigtriangleup B :=(A\setminus B)\cup(B\setminus A)$.
I dont know how to prove this and I dont know where to start.
please give me ...
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
36 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
72 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
51 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
41 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
43 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:
$$C := \{z ...
1
vote
1answer
30 views
Why no cut-vertices or cut edges in a graph where eccentricity is same for all vertices
I need help to prove the following statement.
There are no cut-vertices or cut-edges(bridges) in a graph where eccentricity is same for all vertices. I am getting that if the graph contains a ...
1
vote
1answer
58 views
How do I check the triangle inequality using subsets?
How do I establish the triangle inequality $|x+y| \geq |x|+|y|$ by considering all real numbers as the union of six subsets and checking the inequality on each of those subsets?
0
votes
0answers
33 views
Proof of an intuition for equivalent definitions? (2)
I apologize for the double-post, but I asked the absolute wrong question and received a correct answer for that incorrect question. My apologies.
I have a very strong intuition about the ...
0
votes
1answer
20 views
Proof of an intuition for equivalent definitions?
I have a very strong intuition about the equivalency of the following definitions for Cauchy sequences:
$$\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
994 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
149 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
452 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
41 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
45 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:
$$
a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^n + n + 3 \quad a_0 = 1, a_1 = ...
0
votes
2answers
43 views
Finding this solution to a recurrence relation
So, I know that the recurrence relation $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ has the solution of $a_n = -4(2^n) - n^2 / 4 - 5n / 2 + 1/8 + (39/8)(3^n)$. I just ...
0
votes
1answer
22 views
How to show all solutions for a particular recurrence solution
I've found that the recurrence relation $a_n = 4_{an−1} − 4a_{n−2} + (n + 1)2^n$ has the solution of $an = 2^n(p_0 + p_1n + n^2 + n^3/6)$. I'm just trying to understand the steps necessary to solve ...
2
votes
0answers
234 views
The Dinitz Problem - proof
This theorem is the one that the proof is for
Consider $n^2$ cells arranged in an $(n × n)$-square, and let $(i, j)$
de- note the cell in row $i$ and column $j$. Suppose that for every
cell ...
2
votes
2answers
53 views
Finding the solution to this specific recurrence relation
What would be the solution to $a_n = 7a_{n−2} + 6a_{n−3}$ with $a_0 = 9$,
$a_1 = 10$, and $a_2 = 32$
I can find it for a specific value of (n), but not for just a general solution. Thanks!
1
vote
1answer
53 views
Proof Technique and Factorials
I need to prove that $\;n!+m$ is divisible by $m$ for all integers $n \ge 2$ and $1 \le m \le n$.
1
vote
0answers
35 views
Proving a specific recurrence relation theorem
I'm trying to come up with a proof for this theorem:
Let $c_1$ and $c_2$ be real numbers with $c_2 != 0$. Suppose that $r^2 - c_1 r - c_2 = 0$ has only one root, $r_0$. A sequence $\{a_n\}$ is a ...
1
vote
2answers
44 views
Finding a solution to a recurrence relation
Find the solution to $$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 ...
