Articles of proof writing

Does $(\neg R\to R)\to R$ give rise to a proof strategy?

Take for example proof by contradiction, it can be viewed as a certain deduction in logic which can be used outside of logic to prove many interesting propositions. My question is: can we use $(\neg R\to R)\to R$ as a similar strategy?

Questions of the form “let $G$ be a planar graph such that every face is a…$” Prove that…

Let $G$ be a planar graph with degrees all at least three. Suppose we can draw $G$ in a plane such that every face is a square. Prove the number of vertices of degree three is greater than the number of vertices with degree at least five. Let $D$ be a planar drawing of a […]

Prove by induction that the number of derangements of length $n$ is $D_n = (n-1)(D_{n-1}+D_{n-2}), n>2$

Prove by induction that $$ D_n = (n-1)(D_{n-1}+D_{n-2}), n>2 $$ where $D_n$ is the number of derangements of $n$ objects. I am a little rusty in induction, and here’s what I have done so far. Basis step: This is easy to prove and trivial, it works for $n=3$ and this is the basis step. Induction […]

proof of$\frac{\partial^2 f(x,y)}{\partial x\partial y}$=$\frac{\partial^2 f(x,y)}{\partial y\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\partial x}$$.I saw this relations frequently used in the derivation of different formula for multiple gradient,curl etc. which is fundamental for many other important theorems.My doubt is that is it true for all […]

Proof/intuition that any number can be expressed in binary form and every number will have a unique representation?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation? Clarification: With numbers, I mean whole numbers. Or non negative integers.

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 that the new series is $< 2 \sqrt{x+1}$. Can someone show me the correct process? I failed […]

Prove: If $(g \circ f)$ is bijective, is $f$ bijective?

I need to prove or disprove for a discrete mathematics assignment the following statement: $(g \circ f)$ is bijective $\rightarrow$ $f$ is bijective, $f: X \rightarrow Y$ $\hspace{.5cm} g:Y\rightarrow Z$ All of the domains and codomains here are supposed to be the real numbers. I’m having a hard time understanding how to prove things about […]

Find the limit of function $f_{n}(x) = nx^{n}$

Show that $\lim_{n \rightarrow \infty} nx^{n} = 0$ for $x \in [0, 1), n \in \mathbb{N}$ $\lim_{n \rightarrow \infty} nx^{n}$ $\Rightarrow \lim_{n \rightarrow \infty} n \cdot \lim_{n \rightarrow \infty} x^{n}$ $\Rightarrow \infty * 0$ $\Rightarrow 0$ Is this correct? Now show that $\lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) dx = 1$ $\lim_{n \rightarrow \infty} \int_{0}^{1} nx^{n} […]

The number of vertices in a polytope is finite

This question is an exact duplicate of: Prove that a convex polytope has a finite number of extreme points

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn’t be formal, but in the “usual” theorem-proving style in mathematics) Of course I’ve given it a try and came up with this: Proof: Suppose $\neg \exists x(P(x) \Rightarrow \forall […]