Articles of proof writing

Correct process for proof in graph theory.

I’m working on what I’m sure is a fairly basic proof in graph theory. I must prove that $Every$ $graph$ $G$ $contains$ $a$ $path$ $with$ $\delta(G)$ $edges$. $\delta(G)$ is the minimum degree of the graph $G$. I did ask a very premature question a few days ago about this proof, which was simply a result […]

I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are distinct and x is not among the free variables of L: M[x:=N][y:=L] equals M[y:=L][x:=N[y:=L]] to prove that in the case where […]

Prove that $\displaystyle\int_{x=-1}^{1}P_L(x)P_{L-1}\acute (x)\,\mathrm{d}x=\int_{x=-1}^{1}P_L\acute(x)P_{L+1} (x)\,\mathrm{d}x=0$

A question (Problem $7.4$) in my textbook (Mathematical Methods in the Physical Sciences – 3rd Edition by Mary L. Boas P578) asks me to Use $$\int_{x=-1}^{1}(P_L(x)\cdot\text{any polynomial of degree < L})\,\mathrm{d}x=0\tag{A}$$ to prove that $$\displaystyle\int_{x=-1}^{1}P_L(x)P_{L-1}\acute (x)\,\mathrm{d}x=0\tag{1}$$and gives the hint: $\color{#180}{\fbox{What is the degree of $P_{L−1}(x)$}}$? Also, show that $$\displaystyle\int_{x=-1}^{1}P_L\acute(x)P_{L+1} (x)\,\mathrm{d}x=0\tag{2}$$ Where $P_L(x),P_{L-1}(x)$ represent any general […]

Is this interpretation of the Dirac-measure property correct?

First and foremost, apologies in advance for using an abuse of notation by placing the Dirac measure inside an integral for which I was told that this should not be done from a previous question asked by me. But given the circumstances, I have no choice. This is essentially a word by word copy of […]

Prove that F $ \in \mathbb{R} $ is closed if and only if every Cauchy sequences contained in F has a limit that is also an element of F.

I’m a novice at proofs so I like to write out everything, so please bear with me!. I understand that this is a biconditional statement, and I will have to prove it in the forward and reverse direction. To prove that every Cauchy sequence contained in F has a limit point that is also contained […]

Correctness in Proof By Induction for a Collatz-ish function

I’ve a function that looks like the one mentioned in Collatz Conjecture $$ f(n)= \begin{cases} 1 & \text{if $n=1$}\\ \tfrac12n & \text{if $n \equiv 0 \ \ $ (mod 2)}\\ 3n+1 & \text{if $n \equiv 1 \ $ (mod 2) }\\ \end{cases} \\ , \forall \ \ n \in \mathbb{Z}^+ $$ I want to prove […]

Standard logic notation in mathematics

My profesor is always complaining that my proofs are very long and difficult to read because I never use notation, meaning I say everything in words. Tired of that I decided to study logic by myself and develop my proofs by using the methods of logic. The problem to me now is that I don’t […]

A decomposition of a differentiable function

this time I want to solve this problem: Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be differentiable (may be not in $C^{1}$) and $f(0)=0$, show that there exists $g_{i} : \mathbb{R}^{n} \to \mathbb{R}$ such that: $$f(x)= \sum_{i=1}^{n} x^{i}g_i $$ then, I thought in Taylor decomposition, but the thing is that I can’t use that here because I […]

Proof that when repeatedly splitting a heap of marbles into two and writing down the product of the two heap sizes, the total is ${x \choose 2}$

Here is the problem in full: A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more than 1 marble and form two non-empty heaps from it. One will contain $n$ marbles and the other $m$ […]

How to prove CYK algorithm has $O(n^3)$ running time

I have a final coming up in few days, and the professor mentioned the CYK algorithm. I want to be prepared for the final. I’m trying to find out how to prove the algorithm has worst case running time of $n^3$. Thanks