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.

learn more… | top users | synonyms

2
votes
1answer
107 views

Colimits and epimorphisms.

I am working on a project, and I need to know the proof of this: Any functor which preserves all colimits preserves epimorphisms. So could you please tell me how or where I can find the proof ...
2
votes
1answer
161 views

On how to get a solution for a nonhomogeneous problem for the heat equation.

Evans PDE book presents the following problem (page 87): Write down an explicit formula for a solution of $$\left\{\begin{matrix} u_t-\Delta u+cu=f &\text{ in }\mathbb{R}^n\times(0,\infty) ...
2
votes
3answers
774 views

Monotone and Bounded Sequences: Proof

Say $s_1=1$ and $s_{n+1}=\frac{1}{5}(s_n+7)$ for $n\geq 1$. Prove that the sequence is monotone and bounded, then find the limit.
2
votes
1answer
296 views

Computing Quotient Groups $\mathbb{Z}_4 \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$, $\mathbb{Z} \times \mathbb{Z}_{6}/ \langle (1, 2) \rangle$

Let $G/H = \mathbb{Z}_{4} \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$. I know that $|G/H|$ = 4, so $G/H \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$. Since $G/H$ has an ...
1
vote
6answers
124 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
1
vote
4answers
405 views

Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ

I can do it by induction, thanks to the wonderful people of this website, but I'm not sure how to do it by the Division algorithm. Can anyone help me? I think I can show how 3 divides 2n, but I'm not ...
1
vote
1answer
184 views

Ground plan of Backward direction (<=) - Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$

Apply the identity $p-i \equiv -i \mod p$ for $i=1, \ldots$ to the pink factors $ \begin{align} \color{seagreen}{ (p-1)! } = 1\times 2\times\cdots\times \dfrac{p-1}{2} & \times \quad ...
1
vote
1answer
99 views

Permutation proofs

I have just started going though "An Introduction to the theory of groups" by J.J Rotman. I have questions above the following two exercises: " The identity function $1_X$ on a the set $X$ is a ...
1
vote
5answers
138 views

How to prove that for all natural numbers, $4^n > n^3$?

This is a problem set I have, it's not a homework but it's very important practice... Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably... The ...
1
vote
3answers
108 views

Proof of a trigonometric expression

Let $f(x) = (\sin \frac{πx}{7})^{-1}$. Prove that $f(3) + f(2) = f(1)$. This is another trig question, which I cannot get how to start with. Sum to product identities also did not work.
1
vote
1answer
92 views

How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?

Let $A,B,C$ be sets such that $C\subseteq B$. Let $f: A \to B$ be a function. Prove that $f^{-1} (B\setminus C)=A\setminus f^{-1} (C).$ I really need help with this proof problem. I'm not sure ...
1
vote
3answers
2k views

Hessian matrix of a quadratic form

I need a help with one example. I have to proove that hessian matrix of a quadratic form $f(x)=x^TAx$ is $f^{\prime\prime}(x) = A + A^T$. I am not even sure how the Jaxobian looks like (I never did ...
0
votes
2answers
72 views

Solutions of $\arctan x = 1 - x$. Proofs?

In this question, we examine the equation $\arctan x = 1 - x$. You may assume without proof that $\arctan x$ is continuous on $R$. a) Prove that there is a solution to the equation in the interval ...
0
votes
1answer
63 views

How to prove this form of $n$?

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. From: AOPS Putnam A1 Solution I ...
0
votes
3answers
294 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
0
votes
3answers
124 views

Derive Closed form sum of N^2

Can anyone explain to me how you would derive this ? I have this question asked in a CS class and can't figure out how to derive it. it has to be derived as you would with sum of N ex ...
0
votes
1answer
299 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
12
votes
3answers
2k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) ...
8
votes
1answer
855 views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
7
votes
3answers
252 views

How to understand proof of a limit of a function?

Given the following function: $$ f(x)=\left\{ \begin{array} {cc} 0, & x \text{ irrational, } 0<x<1 \\ \frac{1}{q}, & x=\frac{p}{q} \text{ in lowest terms, } 0<x<1 \end{array} ...
7
votes
5answers
919 views

Constructive proof of boundedness of continuous functions

Consider the theorem for the continuous function: Let $a<b$ be real numbers, and let $f:[a,b]\to{\bf R}$ be a function continuous on $[a,b]$. Then $f$ is a bounded function. The proof in the ...
6
votes
3answers
110 views

How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction

I've done the base step, but how do I prove it is true for $n+1$ without using a fallacy? $$4(n!)>2^{n+2}\quad \text{for } n\geq 4$$ Please help.
5
votes
4answers
514 views

Discrete Math Proofs Involving Real Numbers

I am stuck on these two problems. $1$. Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a}+\frac{1}{b} + \frac{1}{c}) \ge 9$. $2$. Prove that for every three ...
5
votes
3answers
291 views

Help with Cartesian product subsets [duplicate]

I want to prove that if $A \subseteq C\,$ and $\,B \subseteq D,\,$ then $\,A \times B \subseteq C \times D.$ I know that $A \subseteq C \iff a \in A \rightarrow a \in C$ and that $B\subseteq D\iff ...
5
votes
4answers
313 views

Combinatorial Proof

I have trouble coming up with combinatorial proofs. How would you justify this equality? $$ n\binom {n-1}{k-1} = k \binom nk $$
4
votes
3answers
233 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
4
votes
3answers
5k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...
3
votes
1answer
109 views

The number of regions into which a plane is divided by n lines in generic position [duplicate]

Suppose that $n$ lines are drawn on a plane in such a way that no lines are parallel and no three of them intersect at a point. Let $r(n)$ be the number of regions the plane is divided into after ...
3
votes
1answer
55 views

Do absolute convergence of $a_n$ implies convergence of $K_n=\frac{1}{\ln(n^2+1)}\sum_{k=1}^{+\infty}a_k\frac{3k^3-2k}{7-k^3}\sin k$?

The problem asks us to decide whether the following statement is true: Let $\{a_n\}_{n\geq1}$ be any absolutely convergent sequence. Does that imply that the sequence: ...
3
votes
3answers
79 views

Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism

I'm working on the following problem: Let $G$ be a group. Prove that $G$ is abelian if and only if $\varphi(g) = g^2$ is a homomorphism. My solution: First assume that $G$ is an abelian group ...
3
votes
2answers
665 views

Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\sin(2x)=2\sin(x)\cos(x)$?

I was studying math.. and I just realized that I only just memorized these trigonometric equations, but I don't really know the reason behind them. So um... Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and ...
3
votes
2answers
141 views

Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.

This is a recent homework bonus question assigned in my Proofs and Conjectures class. It (evidently) includes and evaluates our understanding of elementary-set theory and how to determine and prove ...
3
votes
2answers
138 views

If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
3
votes
2answers
111 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
3
votes
4answers
108 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
3
votes
1answer
123 views

Proof for recursively defined sets

Language $L\subset \{a,b\}^*$ is such that: $\epsilon \in L$ $a \in L$ For any $x\in L$, $xb\in L$ and $xba\in L$ Nothing else in $L$. Im just learning recursive sets, but with that definition am ...
3
votes
2answers
130 views

Prove through induction that $3^n > n^3$ for $n \geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step. The question is: Use induction to show that $3^n > n^3$ for $n \geq ...
3
votes
2answers
363 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
3
votes
2answers
2k views

Prove the centralizer of an element in group $G$ is a subgroup of $G$

We have a group $G$ where $a$ is an element of $G$. Then we have a set $Z(a) = \{g\in G : ga = ag\}$ called the centralizer of $a$. If I have an $x\in Z(a)$, how do I go about proving that the inverse ...
2
votes
1answer
36 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
2
votes
2answers
80 views

Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [closed]

I tried proving this by contraposition, by saying, "If every number that is being averaged is greater than 7, then the average of a thousand numbers is less than 7." This seems easier to prove, but I ...
2
votes
2answers
554 views

If $x \in\mathbb{Z}$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$

I am learning proofs, and I am stuck with this proposition: Let $x \in\mathbb{Z}$. If $x$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$. I want to use the additive ...
2
votes
2answers
107 views

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction [duplicate]

Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction My proof so far: Let $P(n)$ be $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ Base Case $P(1):$ LHS = $1^3 = 1$ ...
2
votes
2answers
42 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
2
votes
7answers
127 views

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$

Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z.$ I know this is true because any even number that is squared will be even, is it also true than any even number ...
2
votes
2answers
154 views

Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$

I think I have the first part, as I have shown that $q < q + (q'-q)/2 < q'$, but I have trouble in proving that $|q-q'|< \epsilon$. Could someone tell me if there is a better way of showing ...
2
votes
4answers
228 views

Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$ [duplicate]

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b\in \mathbb R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
2
votes
1answer
263 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
2
votes
1answer
118 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
2
votes
2answers
314 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...