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
126 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
41 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 $D_{1,2}f(0,0)$ ...
3
votes
6answers
1k views

How do I begin proving this binomial coefficient identity: ${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$

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
44 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 ...
3
votes
7answers
157 views

Hint in Proving that $n^2\le n!$ [duplicate]

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
1answer
114 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
432 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
83 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
38 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
303 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
1answer
331 views

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

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
38 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
202 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
55 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
47 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
59 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). ...
1
vote
2answers
170 views

Proof by induction that $\sum_{j=0}^n 2^j = 2^{n+1} - 1$

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
82 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"?
2
votes
4answers
173 views

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
52 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
99 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
53 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
45 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
94 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
515 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
136 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
209 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
15 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
19 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 ...
2
votes
2answers
372 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
173 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
42 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
17 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
61 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
100 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
64 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 ...
5
votes
1answer
103 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
42 views

The contrapositive of “if $x$ is even and $x$ is greater than $2$, then there exist prime numbers $p$ and $q$ such that $x = p + q$”

Proposition: If $x$ is even and $x$ is greater than $2$, then there exist prime numbers $p$ and $q$ such that $x = p + q$. Contrapositive: If for all prime numbers $p$ or $q$, $x$ does not equal $p + ...
0
votes
1answer
38 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
70 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
2answers
77 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 ≤ ...
4
votes
6answers
161 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 ...
2
votes
0answers
144 views

Power Set, Bijection Function, Equivalence Relation [duplicate]

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
40 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 & ...
1
vote
1answer
33 views

basic conditional probability proof

I having trouble with the following proof: $$P((A \cap B) \mid B) = P(A\mid B).$$ I get that $P(A\mid B) = P(A \cap B) / P (B)$, but I am unsure of how to proceed.
4
votes
0answers
154 views

Prove $f_\infty: A_\infty \rightarrow B_\infty$ is a bijection

Update: I was given some hints at how to approach this problem $A_\infty $ and $B_\infty$ are sets, not maps. (which is strange because there are function definitions coming into play here) The ...
5
votes
2answers
211 views

Is (the proof of) Fermat's last theorem completely, utterly, totally accepted like $3+4=7$?

If a mathematician would/does make use of Fermat's last theorem in a proof in a publication, would s/he still make use of some kind of caveat, like: "assuming Fermat's last theorem is true" or ...
0
votes
2answers
52 views

Sum of the eigenvalues

if $V$ is a finite-dimensional vector space and $t \in \mathcal L (V,V) $is such that $t^2 = id_V$ prove that the sum of eigenvalues of t is an integer. I started the prove as such: Let $\lambda_1 ...
2
votes
1answer
40 views

Show that $R \cap R^*$ and $R \cup R^*$ are equivalence relations.

Let $R$ be a reflexive and transitive relation on a set $S$. Let $R^*$ be the dual relation, $(a,b) \in R^*$ if and only if $(b,a) \in R$. Show that $R \cap R^*$ and $R \cup R^*$ are equivalence ...
0
votes
4answers
57 views

Help with a certain proof

For all $x,y \in \mathbb{R} - \{0\}$, $(xy)^{-1}=x^{-1}y^{-1}$. I was wondering how I could solve this.