Recent content by TyroneTheDino

A subset E of X is called convex if, for any ##x,y \in E## and ##t \in (0,1)## then ##(1-t)x + ty \in E##. So by the inequality I wrote since ##\alpha f(x) + (1-\alpha)f(y)## is contained in the set it is convex?

Homework Statement [/B] Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex. Homework Equations [/B] If f is convex...

Okay, I take from this because \sqrt{x_1}= \sqrt{x_2}, \sqrt{x_1}^2= \sqrt{x_2}^2, so x1=x2. So this function is one to one because I can prove that . Correct?

Homework Statement I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}## ##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one Homework EquationsThe Attempt at a Solution I know for certain that this function is not onto given the codomain of real...

I updated it to define An.

Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...

Ah I understand, this makes more sense to me now. Thank you.

But one question: since i had ##\neg[(p\mid q)=x]## when the negation moves inside the expression it becomes: ##[(p\nmid q )\ne x]## Correct?

Oh thank you, correction made.

Homework Statement Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers Homework EquationsThe Attempt at a Solution I believe the existence of rational numbers can be stated as: ##(\forall n \in \Re)(\exists p,q \in...

Homework Statement Express the following statement using only quantifiers. (You may only use the set of Real and Natural Numbers) 1. There is no largest irrational number. Homework Equations ##\forall=## for all ##\exists##=there exists The Attempt at a Solution I express the existence of...

Homework Statement Simplify the following statement as much as you can: (b). ##(3<4) \wedge (3<6)## Homework Equations ##\wedge= and## The Attempt at a Solution I figured that I could just write this as ##3<4<6##, but then I considered what if I didn't know that ##4<6## If it was just...

How about since ##\frac{x^2}{a^2}+\frac{y^2}{b^2}=1## ##(x-1)^2+y^2=1## Can I set each side equal to each other or should I solve for y^2 of the circle equation to plug into the ellipse equation.

I know this problem is emulative of https://www.physicsforums.com/threads/optimization-minimize-area-of-an-ellipse-enclosing-a-circle.270437/ this one however I am just getting into multivariable differentiation so this is very confusing to me.

I don't think I'm following that relationship either. How am I supposed to know the relationship between the ellipse and the circle.