For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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3
votes
1answer
106 views

Proof the following trig series

Prove that $$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$ I am not necessarily looking for a ...
6
votes
2answers
426 views

Real Numbers is a subset of Complex Numbers?

So, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$ But, since the complex numbers' definition is $\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}$, doesn't that mean ...
1
vote
1answer
58 views

Combinatorics identity sum of

Prove that: $$\sum^{n}_{k=0}\binom{k}{2n-k}2^k = 2^{2n}$$ By using only combinatorics identities.
0
votes
2answers
62 views

prove $f^{-1}(B)=A$

I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$ I want to prove that $f^{-1}(B)=A$. I just assume that here one is talking about ...
3
votes
2answers
140 views

Where is wrong in this proof [duplicate]

Suppose $a=b$. Multiplying by $a$ on both sides gives $a^2 = ab$. Then we subtract $b^2$ on both sides, and get $a^2-b^2 = ab-b^2$. Obviously, $(a-b)(a+b) = b(a-b)$, so dividing by $a - b$, we find ...
2
votes
1answer
50 views

Residue of a 1-form in a Riemann Surface does not depend of the chart

Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by $$ ...
0
votes
1answer
18 views

Clues to prove average in T is minor or equal than average in a smaller inner interval.

Suppose I want to prove (or disprove) this assertion Let $f$ be a discrete function, $T,h,k$ are constants So these terms are averages over $T$ and over $h$ $\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
14
votes
4answers
1k views

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$ $(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
0
votes
0answers
91 views

Proof contraction differentiable function

$g$ : $R$ $\rightarrow$ $R$ be a diferentiable function such that $-1$ < $a$ < $b$ < $0$ where for $y$ $\in$ $\Re$, $a$ $\le$ $g'(t)$ $\le$ $b $ Prove that $g(t) = t + f(t)$ is a contraction ...
1
vote
0answers
106 views

Proving that the circumcenter is the centroid

Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
2
votes
2answers
1k views

Help with proofs: Show that $AA^T$ and $A^TA$ are symmetric

I need help with a proof for my liner algebra class. If $A$ is a square matrix, then $AA^T$ and $A^TA$ are symmetric. I have no idea where to start!
3
votes
1answer
85 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
4
votes
3answers
309 views

Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]

The problem is following, prove that: $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$ I've tried solving this problem using mathematical induction, but I ...
1
vote
2answers
425 views

proof by induction to demonstrate all even Fibonacci numbers have indices divisible by 3

I am practicing proof by induction, and would like to use induction to prove the following hypothesis about the Fibonacci numbers: $$(\forall n\ge0) \space 0\equiv n\space mod \space 3 \iff 0 \equiv ...
1
vote
1answer
46 views

$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$

Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
11
votes
4answers
294 views

Which Cross Product for the Desired Orientation of a Sphere ? [Stewart P1091 16.7.23]

P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation. P1087: ...
2
votes
2answers
2k views

How do I prove the arithmetic-geometric mean inequality?

I am following along with this bare-bones proof of the arithmetic-geometric mean inequality with two real numbers. I'm having difficulty understanding the logic behind this step: $$ ...
3
votes
1answer
93 views

Rice's theorem_Theory of computation

Is there any body tell me, where is wrong in this proof Problem: The set of number of turing machine that has 5 state is decidable or not? Answer: The set is obviously 'Set of partial computable ...
6
votes
3answers
182 views

Extreme Value Theorem Proof (Spivak)

Them: If $f$ is continuous on $[a,b]$, then there is a $y$ in $[a,b]$ such that $f(y) \geq f(x)$ for each $x \in [a,b]$ Proof. We already know that $f$ is bounded on $[a,b]$, which means that ...
1
vote
1answer
52 views

Is this proof on the product of $X$ OK?

Let $X^2$ be star $\sigma$-compact and $F$ be a closed subset in $X^2$. If $\mathcal{U}$ is an open cover of $F$, then there exists a $\sigma$-compact subset $A$ of $X$, such that $F \subseteq ...
1
vote
1answer
74 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
2
votes
1answer
62 views

Are there examples of theorems proved via proper (i.e. non-conservative) extensions?

This is not a question about set theory specifically, but lets talk about ZFC just for concreteness Suppose we have a sentence $\phi$ in the language of ZFC, and a proof that either $(\mathrm{ZFC} ...
0
votes
1answer
71 views

Finding a reccurence relation for the following problem

A circular disk is cut into n distint sectors, each shaped liek a piece of pie and all meeting at the center point of the disk. Each sector is to be painted red, green, yellow, or blue in such a way ...
-2
votes
1answer
142 views

Given the following recurrence relation, prove using mathematical induction

How can we prove this using mathematical induction? $m_1 = 0$ $m_k = m_{\lfloor (k/2) \rfloor} + m_{\lceil (k/2) \rceil} + k-1$ for all integers $k \geq 1$ Prove using mathematical induction that ...
0
votes
6answers
2k views

Finding the number of subsets of S

How can we find the number of subsets of $S=\{1,2,3,...,10\}$ that contain neither 5 nor 6? Thanks!
1
vote
2answers
133 views

Use the binomial theorem to expand

How can we expand this using the binomial theorem? $(x^2 + 1/x)^7$
7
votes
2answers
433 views

How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
1
vote
2answers
145 views

A typo in Spivak's solution?

Problem Solution I honestly cannot figure out what he is doing. On one hand, I think Spivak wants to write $|\phi(b)/b^n| > 1/2$ instead of $|\phi(b)/b^2| < 1/2$. On the other ...
1
vote
2answers
132 views

Induction on the Fibonacci sequence?

Prove by induction that the $i$th Fibonacci number satisfies the equality: $$F_i = \frac {\phi^i - \hat\phi{}^i}{\sqrt5}$$ where $\phi$ is the golden ratio and $\hat\phi$ is its conjugate. ...
3
votes
2answers
48 views

Show that $\exists A \subset \mathbb{R}$ such that $\forall x$ $\in \mathbb{R}$, we may write $x$ uniquely as $x=a+q$, where $a\in A,q\in\mathbb{Q}$.

Not sure where to go with this one. Clearly will have to use the axiom of choice at some point. I haven't been able to think of a good example for the set A. Once we've got that, it'd be a matter of ...
1
vote
0answers
30 views

Farthest vector pair in subset of unit circle.

This question is extended from this question Given a set $S$ and a pair of vector $x,y\in S$ In this version the set $S$ is a subset of unit circle. That is for all $s \in S$, $||s||=1$ Does the ...
1
vote
1answer
23 views

Showing a pair of vector is the farthest vector pair in certain set

Given a set $S$ and a pair of vector $x,y\in S$ I would like to show $x$ and $y$ are the farthest vector pair in the set $S$ I start with showing there doesn't exist a vector $a \in S$ s.t. ...
2
votes
3answers
85 views

Prove that $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$.

Logically, the following proposition makes sense: $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$ Or, in english, if sets $S$ and $T$ share no elements, and the union of ...
5
votes
2answers
619 views

How can a matrix be Hermitian, unitary, and diagonal all at once

I was given the following problem in class, and I'm not really sure how to begin this proof. Describe all 3 by 3 matrices that are simultaneously Hermitian, unitary, and diagonal. How many are ...
14
votes
1answer
536 views

Spivak's proof that every polynomial of odd degree has a root

I have the second edition of Spivak. Consider Can someone tell me why he considers $2n|a_{n-1}| \dots$? Later he shows everything is squeezed between -1/2 and 1/2 and he gets the desired result. I ...
1
vote
1answer
49 views

Let $X$ has countable extent. Does $X^2$ have countable extent?

Definition 1: A space $X$ has countable extent if every uncountable subset of $X$ has a limit point in $X$. I'm struggling with this question: Question 2: Let $X$ has countable extent. Does ...
0
votes
1answer
215 views

Lie derivative: Leibniz rule proof

How can I prove $\mathcal{L}_v(\omega\wedge\alpha) = (\mathcal{L}_v\omega)\wedge\alpha + \omega\wedge(\mathcal{L}_v\alpha)$ ?
4
votes
0answers
91 views

Infinite “String” of Implication Statements

This question is inspired by the conversations at Does this require transfinite induction? First of all, does an infinite string of implication statements have a conclusion? I don't think so, but I ...
2
votes
2answers
834 views

Proving Cauchy's Mean Value Theorem

In proving Cauchy's mean value theorem, the first step is to use this function: $$ h(x)=[f(b)−f(a)]g(x)−[g(b)−g(a)]f(x)$$ I've seen this in many textbooks but none of them actually show how they got ...
-1
votes
1answer
1k views

Proof of divisibility by 2 and 3 if and only if divisible by 6

I can't find a way of proving that: For integer a, a is divisible by 2 and divisible by 3 if and only if a is divisible by 6. I’m not sure where to go from here. Any help would be great!
2
votes
3answers
298 views

Prove that Statements forms are tautologies

Given variable statement forms $A$ and $B$. How to prove that if $(A\land B)$ is a tautology then $A$ and $B$ are tautologies too?. Mi approach would be a proof by contradiction, something like: If ...
3
votes
1answer
151 views

Prove that there exist infinitely many squares $a$ such that $\sqrt{\sqrt{a}}$ is a square

I was just thinking about squares while randomly punched numbers into my calculator and I was wondering do there exist infinitely many squares such that $\sqrt{\sqrt{a}}$ is a square and $a$ is also a ...
11
votes
1answer
270 views

Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$

How to find all possible functions $f(x,y)$ such as: $$ \frac{\sqrt{3}}{2}f_x+\frac{1}{2}f_y=0$$ (with $f_x = \frac{\partial{f}}{\partial{x}}$ ) Here's everything I tried: 1) I can guess the ...
2
votes
2answers
82 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 ...
5
votes
3answers
239 views

Pythagorean theorem and its cause

I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the ...
1
vote
2answers
83 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 ...
1
vote
2answers
75 views

$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$

Question Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$? Attempt \begin{eqnarray} T\circ ...
2
votes
1answer
71 views

If $x,y$ are elements of $\mathbb{R}$ and $x>0$ then there is a positive integer $n$ s.t. $nx > y$

Im reading a proof about this The proof is here. Let $A$ be the set of all $nx$, where $n$ runs through the positive integers. If $nx \le y$, then $y$ would be an upper bound of $A$. (start ...
7
votes
1answer
353 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus ...
-1
votes
1answer
61 views

Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$

Theorem to prove: Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...