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
2answers
108 views

Combinatorial Proof For Counting All Binary Strings

The Question Provide a combinatorial proof for the following: For $n \ge 1$, $$2^n = \binom{n+1}1+\binom{n+1}3+\ldots+\begin{cases} \binom{n+1}n,&\text{if }n\text{ is odd}\\\\ ...
4
votes
3answers
64 views

The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.

I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
2
votes
1answer
38 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
0
votes
1answer
30 views

Proof Using iff Intermediate Lines

I am posting this question motivated by Bungo's response to my question here -- scroll down to his/her response and comment. It was the first time I've seen this technique. It looks like a circular ...
0
votes
2answers
43 views

How would one prove this flavour of the triangle inequality?

I have to prove $|z_1 - z_2| \leq |z_1|+|z_2|$ where $z_1,z_2$ are in $\mathbb{C}$. What I wrote down is: $$|z_1| = |z_1+z_2-z_2| \geq |z_1-z_2|-|z_2|\implies |z_1|+|z_2|\geq |z_1-z_2|,$$as desired. ...
0
votes
0answers
25 views

To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
0
votes
2answers
64 views

Proof of a point's existence in an open interval

Well let us begin consider a set $A$ $$A = \{a \le x \le b \space | \space f(x) > 0 \}$$ Lets take $\alpha = \sup A$ and $\beta = \inf A$ What we must do is prove that $\alpha = d$ and ...
1
vote
1answer
20 views

Next step to take in direct proof for one to one?

This is from Discrete Mathematics and its Applications And the definition of strictly increasing. Here is my work so far. I know that a direct proof involves making an assumption p, which in ...
2
votes
1answer
31 views

Let A1,A2,…,An be distinct subsets of a set X. Then there is subset Y with size <=n-1, s.t. all intersections are all distinct.

Let $A_1,A_2,\dotsc,A_n$ be distinct subsets of a set $X$. Then there is subset $Y$ with size $\le n-1$, s.t. all intersections of $A_i$ with $Y$ are all distinct. I am trying to prove it with ...
0
votes
0answers
21 views

Is the $T$-annihilator of the sum of vectors the product of the $T$-annihilators of the vectors?

Let $T:V\to V$ be a linear operator on a finite dimensional vector space and let $p_1(x)$ and $p_2(x)$ be the T-annihilators of $\vec v_1, \vec v_2 \in V$ respectively (that is $p_1(x)$ and $p_2(x)$ ...
10
votes
4answers
374 views

How can I complete this proof by contradiction?

This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers $x$ and $y$ to the equation $2x^2 + 5y^2 = 14$. I am trying to use proof by ...
1
vote
2answers
53 views

Help understanding what is asked of me

Is there a "min(a,b)" in math I do not know about? PLEASE DO NOT ANSWER THE QUESTION. I just need to know about "min" and what it stands for so I can figure out the question. Question: Use a ...
1
vote
4answers
48 views

A Proof that Orthogonal Complement is unique

So our professor asked us to prove that considering any subspace $S$ of a vector space $V$, the orthogonal complement $S^{\perp}$ is unique. I have devised a proof and I am not sure whether this ...
2
votes
4answers
74 views

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can't find a way to manipulate this expression and find ...
2
votes
2answers
71 views

Proving by induction that any two natural numbers are equal.

This is something I've been working on for a while now; although it seems trivial, I am confused. I can't seem to find the error. Originally I thought the problem was with the base case, then I ...
0
votes
2answers
56 views

Proving that there is an irrational number between any two unequal rational numbers.

I'm trying to prove that there is an irrational number between any two unequal rational numbers $a, b$. Here's a "proof" I have right now, but I'm not sure if it works. Let $a, b$ be two unequal ...
2
votes
3answers
61 views

How can I prove a limit is infinity?

How can I prove that the $\lim \limits_{x \to 1^+} \frac{x^2}{x-1}=\infty$ using the $\epsilon -\delta$ definition of a limit? I think I start with $\forall$M>0, want $\delta$>0. After that I'm not ...
2
votes
2answers
25 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 ...
1
vote
4answers
38 views

Using proof by contraposition to show that if $n\in\mathbb Z$ and $3n+2$ is even, then $n$ is even

I have my answer below but there is one step that I am not understanding...and maybe my brain is just not trained to understand it. Prove that if $n$ is an integer and $3n+2$ is even, then $n$ is ...
0
votes
2answers
43 views

How to prove that the Nested Interval Theorem fails to hold in $\mathbb Q$?

Claim: The Nested Interval Theorem does not hold in $\mathbb Q$. I can prove this by using sequences $a_n$ and $b_n$ where $a_n < b_n$ and they both converge for an $x$ which is any irrational ...
1
vote
1answer
11 views

Proof of a tree with a vertex of degree k and less than k vertices of degree 1

The question is : Does there exist a tree with a vertex of degree k and less than k vertices of degree 1? I tried a lot but it is impossible to find. There is no tree with a vertex degree k and less ...
1
vote
1answer
43 views

Proof of Fundamental Lemma of Calculus of Variations

Let me preface this question by saying I'm actually a physicist, not a mathematician, so a lot of the language I see you guys using here is over my head, so if you can keep it simple, that would be ...
0
votes
2answers
40 views

Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and ...
0
votes
0answers
51 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
0
votes
4answers
97 views

Why can't I prove this statement by simple induction? Sum of $1/2^1 + \cdots+ n/2^n = x$

I have to prove the following: $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{2 + n}{2^n}. $$ I am trying to prove this by simple induction. First, I proved that $P(1)$ ...
1
vote
2answers
23 views

How to prove $\omega$ bound without using limit?

How to show $n^{3.4} - 2015n^{2} + 3$ $\in$ $\omega(n^{3})$ without using limit? According to the definition of $\omega$, $f(n)$ $\in$ $\omega(g(n))$ if and only if $\forall c > 0$, $\exists n_0$ ...
1
vote
1answer
48 views

Prove $\int $ $(1/x)$ dx = $ln|x| + c$.

Note the domain we are working with is $x$ is all real numbers except $0$. My solution: Separate the question into two cases. (1) Prove that the left-side and the right-side of the equation are ...
1
vote
2answers
42 views

Prove $\mid\frac{2a}{b} + \frac{2b}{a} \mid \ge 4, \forall a,b \in \mathbb{R} - \left\{0\right\}$

I started by squaring both sides and proving: $(\mid\frac{2a}{b} + \frac{2b}{a} \mid)^2 \ge 4^2, \forall a,b \in \mathbb{R} - \left\{0\right\}$ My work: Consider: $(\mid\frac{2a}{b} + \frac{2b}{a} ...
1
vote
1answer
39 views

How to proove the following general form of proof

Suppose I have a statement $p(m,n)$ where $m,n \in \mathbb{N}$ that I want to proove. Suppose I have proofs of the following: $p(1,n)$ true for all $n \in \mathbb{N}$. $p(m,1)$ true for all $m \in ...
2
votes
1answer
38 views

$4x^2+1$ factors only into $4y+1$ primes

How can one prove that numbers of the form $4x^2+1$ can only be divided by primes of the form $4y+1$ (e.g. there is no $x$ for which $7$ divides $4x^2+1$)? On a quick lookup, the statement is given ...
2
votes
1answer
33 views

Is $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ bijective and find the inverse function!

$F$ is the set of the sequences in $\mathbb C$ and $R$ is the set of the series in $\mathbb C$. $f: F \to R, \ (a_j)_{j \in \mathbb N} \mapsto \sum_{j \in \mathbb N} \ a_j $ Now $\sum_{j \in \mathbb ...
0
votes
1answer
25 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
1
vote
0answers
29 views

Complex number - prove an inequality

Question: Given that:$$z^n\tan\theta_0 + z^{n-1}\tan\theta_1 + z^{n-2}\tan\theta_2 + ... + \tan\theta_n = 3$$ And that $\theta_i \in (0, \frac{\pi}{4})$, prove that: $$|z| > \frac{2}{3}$$ ...
2
votes
1answer
41 views

Complex numbers - minimum value proof

Question: For:$$|z - z_1|^2+|z - z_2|^2+|z - z_3|^2+\cdots+|z - z_n|^2 = S$$ Prove that the minimum value of $S$ is when:$$z = \frac{z_1+z_2+z_3+\cdots+z_n}{n}$$ I have no idea how to even ...
0
votes
1answer
36 views

Base 10 proof strategy

Let $a$ be a number written (in base 10) as $$a=a_0\cdot10^0+a_1\cdot10^1+a_2\cdot10^2+\cdots+a_n\cdot10^n$$ where $0\leq a_i <10$. Prove the following: 2 divides $a$ if and only if 2 ...
1
vote
2answers
61 views

How tot start proving $A \times B \times C \ne (A \times B) \times C$?

This is a problem from Discrete Mathematics and its Applications: Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same. I understand the process behind the ...
4
votes
2answers
107 views

Second Grade Homework Problem - Methodology

Ashamed to admit that I cannot aid my friend's niece with her second grade homework problem. So much for that college education, eh? Here's the problem. Using only the natural numbers 1 through 9 ...
0
votes
1answer
52 views

Property of continuous functions regarding maximum

Claim 1: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ assumes a maximum value I know there's a theorem that states if $f$ is a continuous real-valued function on a closed interval ...
0
votes
1answer
58 views

Proof strategy for simple proofs.

I'm currently in a discrete mathematics course and I'm having quite a bit of trouble with the idea of proofs. From what I understand the one I've been stuck on is also rather simple but to me it's ...
3
votes
1answer
32 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
4
votes
0answers
73 views

Is $\frac{\pi}{e}$ an algebraic integer?

From what I know, it is still an open question whether or not $\frac{\pi}{e}$ is irrational, but is there a proof that $\frac{\pi}{e}$ is not an algebraic integer?
0
votes
0answers
19 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
0
votes
4answers
31 views

Proof that composition of invertible linear transformations is invertible (without determinants)

A crucial concept in linear algebra is that the composition of two invertible linear transformations is itself invertible. Here is the first proof I learned of this fact: Proof: Suppose that $T_1: ...
2
votes
2answers
56 views

Closed form solution and combinatorial proof.

First of all, I would like to figure out a closed form solution for the following summation: $$\sum^{n}_{k=0} C(n,k)\cdot C(2n,n+k)$$ Where C(n,k) means n choose k, or $\frac{n!}{(n-k)!\cdot k!}$ ...
2
votes
1answer
28 views

Suppose $R$ is partial order, prove that $R^{-1}$ is also a partial order

Suppose $R$ is partial order, prove that $R^{-1}$ is also a partial order. A partial order is a binary relation that is reflexive, anti-symmetric and transitive. So if $R$ is a partial order, ...
0
votes
1answer
51 views

Doubt : Invariance in Geometry

I was working my way through some Proof Problems in Discrete Maths by Rosen, when I came across the following question: What Geometric proposition ( having an invariant property ) does this ...
2
votes
2answers
91 views

If n^2 is even n is even

I understand that there are already several answers to how to prove this question: Prove if $n^2$ is even, then $n$ is even. Prove that if $n^2$ is even then $n$ is even I am trying to understand ...
2
votes
1answer
47 views

Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
2
votes
2answers
49 views

Discrete Math Proof: $A \cup B$

I'm preparing ahead for a Discrete Math course coming up this year by doing some practice problems supplemented by online notes. The problem I'm having trouble proving is the following: $A \cup B ...
0
votes
2answers
37 views

What to use for r in proof by contradiction?

This is a problem from Discrete Mathematics and its applications To this proof, I am trying to use proof by contradiction. Here is how the book described the process of proof by contradiction. I ...