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.

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0
votes
2answers
27 views

Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
0
votes
2answers
16 views

Show that $f$ is everywhere differentiable and the partials commute

Take the function $$ f(x,y) = \begin{cases}\frac{x^3y -xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}. $$ Show that it is everywhere differentiable and that $D_{1,2}f(0,0)$ ...
2
votes
7answers
164 views

How do I begin proving this binomial coefficient identity?

This is a homework question. I'm asked to prove the identity: $${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$$ (The sum ends with ${n\choose n} = 1$, with the sign of the ...
1
vote
1answer
33 views

Uncountability of a nonmeasurable set

As per the Vitali's theorem, every measurable set of positive measure has a subset which is nonmeasurable. Which proceeds by defining a rational equivalence, followed by using the axiom of choice on ...
0
votes
0answers
7 views

Let n be a natural number. Any set, {a1,a2,…,an}, of n integers for which no two are congruent modulo n is a complete residue system modulo n.

I'm not very good at number theory but it seems to me that this can just simply proved by the definition of a complete residue system. Is it that simple?
1
vote
5answers
75 views

Hint in Proving that $n^2\le n!$

The problem is to find the values of n such $n^2\le n!$. I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$} I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...
1
vote
0answers
14 views

Proof: Convex set of a quadrilateral is a convex quadrilateral

Prove that $\Box ABCD$ is a convex set whenever $\Box ABCD$ is a convex quadrilateral. Things I know: A set of points $S$ is said to be a convex set if for every pair of points $A$ and $B$ in $S$, ...
2
votes
1answer
29 views

Spanning trees in planar dual graph

The amount of spanning trees in a planar graph G is equal to the amount of spanning trees in the dual graph G*. I would like to proove this, i know it's true, but i would like to show that it holds ...
1
vote
1answer
31 views

Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
0
votes
1answer
21 views

Eisenstein's criterion pf

I know that 'Eisenstein's criterion'. I know that pf of state "(NOT $p$|$a_{n}$), [$P|a_i$ for ($0\le i\le n-1$)], (NOT $p^2$|$a_0$)". I know regular way. but I hope to Second pf way. $\;$ $\;$ ...
1
vote
1answer
13 views

commutative ring and unity elements proof

So this is a review problem in our book I came across and i really want to understand it but I am just not having any luck, I did some research and found a guide on solving it but that's not really ...
2
votes
2answers
54 views

Prove $\gcd(ka,kb) = k*\gcd(a,b)$

For all $k > 0,\ k\in \Bbb Z$ . Prove $$\gcd(k*a,\ k*b) = k *\gcd(a,\ b)$$ I think I understand what this wants but I can't figure out how to set up a formal proof. These are the guidelines we ...
-1
votes
2answers
23 views

prove: (a|b*c) ^ (gcd(a,b)=1) implies a|c [duplicate]

i need help with the following prove: (a|bc) ^ (gcd(a,b)=1) implies a|c following these writing guidelines http://i.imgur.com/qpIYqPp.png What I know so far: By the Euclidean algorithm there are ...
1
vote
1answer
16 views

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W This is something from a practice sheet I got. I'm studying for a linear algebra final. I am unsure if we have ...
2
votes
1answer
25 views

Doubt on proof of Implicit function theorem

On The second part of the proof, where it's stated that V is open as it is the inverse image of the open set $V_0$ under the continuous mapping $y \rightarrow (0, y)$. Let $\pi$ be this continuous ...
0
votes
1answer
32 views

$f(x)$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$

show that $f(x) (\in \Bbb Z[x])$ is irreducible and primitive over $\Bbb Q$, then $f(x)$ is irreducible over $\Bbb Z$. How pf it? I tried it. MY pf) Suppose that $f(x)$ is reducible over $\Bbb Z$. ...
0
votes
1answer
43 views

Why is P or not P is unsatisifiable by construction?

A proof of predicate logic inability to express graph reachibility (page 63) involves a formula which can be interpreted as (there is no path, no matter what is the length) or (there is some path). ...
0
votes
2answers
29 views

Proof by induction sum $2^j = 2^{n+1} - 1$

I am trying to solve a previous test for an exam, and there are no solutions. The problem I am trying to solve is If $n$ is a natural number, then $1 + 2 + 2^2 + 2+3 + ... + 2^n = 2^{n+1} -1$ ...
1
vote
2answers
70 views

Are there any proofs that use $0/0$ is indeterminate?

I'd imagine that "$n/0$ is undefined $\forall n\neq 0$" is very useful in finding contradictions, but are there any proofs that somehow use "$0/0$ is indeterminate"?
0
votes
0answers
10 views

Showing that s'm is a common multiple of m and n

so in class teacher gave us this algorithm GCD(m,n)=GCD(n mod m, m). after that we used it to find s and t. for example we found GCD of 453 and 174 and their s and t by making a table like this ...
0
votes
2answers
52 views
+50

Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
0
votes
2answers
23 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
3
votes
0answers
33 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
1
vote
1answer
40 views

Prove for all $ n \in N,gcd(2n+1,9n+4)=1$

Question: Prove for all $ n \in N,gcd(2n+1,9n+4)=1$ Attempt: I want to use Euclid's Algorithm because it seemed to be easier than what my book was doing which was manually finding the linear ...
1
vote
1answer
36 views

Is there a more concise way?

I am currently trying to prove Taylor's Theorem, this is the proof I am given: I am finding this very long and hard to follow, can anyone give me a more concise, clear proof? Thanks
0
votes
3answers
64 views

Having trouble understanding Cantors proof that real numbers are uncountable

I found this video very easy to follow and understood the proof. https://www.youtube.com/watch?v=mEEM_dLWY0g However, I am still having trouble understanding the proof presented to me in my csmath ...
1
vote
1answer
43 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
1
vote
3answers
65 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
0
votes
3answers
12 views

conditional Probability proof with inequalities

Let A, B be events taken from a sample space Ω (with Pr(A) > 0 and Pr(B) > 0). If Pr(B|A) < Pr(B), prove that Pr(A|B) < Pr(A). I am a bit confused with this one. Any help would be appreciated. ...
0
votes
2answers
13 views

Proof that in any base $b$, the result of multipling two numbers of $k$ digits, doesn't recuire more than $2k$ digits

The proof that I came up whit is: Let, $c$ be $b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k r_k$ and $d = b^0 r_0'+ b^1 r_1'+b^2 r_2'+...+b^k r_k'$ then multipling both: $$(b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k ...
0
votes
2answers
16 views

Proof that in any base, the sum of two numbers of fixed precision can't have carry more than 1

The best proof that I came with is, given any base $b$, let $c$ be the greatest number can be written whit $n$ digits. Then the number will be: $$c=b^0(b-1)+b^1(b-1)+\cdots +b^n(b-1)$$ Summing this ...
-1
votes
0answers
20 views

Suppose f:A to B. Suppose that C subset A and D subset B. Prove or give a counter example: f (c) subset D iff C subset f^-1(D) [closed]

Suppose f:A to B. Suppose that C subset A and D subset B. Prove or give a counter example: f (c) subset C iff C subset f^-1(D)
2
votes
2answers
34 views

Proving corollary to Euler's formula by induction

I'm currently looking at two proofs to the following corollary to Euler's formula and I'm not quite seeing how the authors can make a specific assumption in their proof. One proof comes from my ...
1
vote
1answer
38 views

Prove that if graph $G$ is a 3-connected planar graph then its dual must be simple.

I'm trying to study for a quiz. I think I'm on the right track with this problem, however, I'm having a difficult time formalizing it. Prove that if graph $G$ is a 3-connected planar graph then its ...
2
votes
1answer
32 views

What does $g(f(x))=x$ imply?

Let $f: X\rightarrow Y$ and $g:Y\rightarrow X$ be functions such that $g(f(x))=x$ for all $x\in X$. (a) Prove that $f$ is injective. (b) Prove that $g$ is surjective. (c) Give an example of a pair ...
1
vote
0answers
15 views

If $X$ is a finite set of cardinality $n$, where $n$ exists in $P$, show that the following conditions on a function $f: X \to X$ are equivalent: [duplicate]

(a) $f$ is an injection (b) $f$ is a surjection (c) $f$ is a bijection I know that (c) implies (a) and (b) and (a) and (b) imply (c). I also have the following definition that I've been playing ...
1
vote
2answers
31 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
0
votes
4answers
60 views

Prove that (integer + non-integer) never equals an integer.

My question is how do you prove that given an integer $x$ and a number $y$, the only way for $x + y$ to be an integer is if $y$ is also an integer. I can see how to prove by induction that an integer ...
2
votes
1answer
31 views

Showing that if derivative is 0, function is constant ($f: U \rightarrow \mathbb{R}$ where $U \subset \mathbb{R}^n$)

Here's the question: Suppose that $f: U \rightarrow \mathbb{R}$ is differentiable on the open subset $U\subset \mathbb{R}^n$, and $Df(x) =0$ for all $x\in U$. Show that $f$ is constant on $U$. My ...
0
votes
0answers
8 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
5
votes
1answer
79 views

For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer

I'm trying to prove this by induction but seem to be getting nowhere.
-2
votes
1answer
28 views
0
votes
1answer
24 views

Is a cascaded chaotic system is still chaotic?

I am curious whether a new system which cascades two individual chaotic systems is always chaotic. My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that $${\rm ...
0
votes
1answer
35 views

Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
0
votes
1answer
41 views

General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
3
votes
6answers
74 views

Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$

How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$ for any two integers $a$ and $b$? Intuitively it is true because when you divide $a$ and $b$ by ...
0
votes
0answers
68 views

Prove that a certain graph and its dual are 4-colorable

Let $G$ be a simple planar graph with fewer than 12 faces. Suppose that each vertex of $G$ has degree at least $3$. prove that $G$ and its dual are 4-colorable. I'm not too sure how to approach ...
0
votes
0answers
6 views

Determining the minimal number of axis to test against in the SAT (Separating Axis Theorem)

Most implementations of the SAT algorithm I've seen involve testing each axis in either shape being tested against for collisions. But I recently implemented the SAT algorithm in python and noticed ...
1
vote
0answers
48 views

Power Set, Bijection Function, Equivalence Relation

Let $S$ be a set and $P(S)$ the power set of $S$. For sets $A,B⊆P(S)$, we say that $A \sim B$ if there exists a bijective function $f: A \rightarrow B$. Show that $\sim $ is an equivalence relation.
0
votes
2answers
23 views

discrete fourier transform proof (show equals n*I)

Let $w=e^{(-2\pi i/n)}$. Let $W$ be an $n \times n$ matrix defined by $$ W = \begin{pmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & w & w^2 & w^3 & \cdots & ...