Difficult logic question with conditional statements [click here for image of question][1]      [1]:       The question states: in this problem we will only consider (real- valued) function with domain $\mathbb{R}$ (real numbers). We define two new concepts. Let $f$ and $g$ be two functions;         we say $f$ is a rival of $g$ if $∃x∈ \mathbb{R}, s.t. ∀y∈ \mathbb{R}, x&lt; y \text{ then } |f(x)-g(x)|&lt;|f(y)-g(y)|$   we say $f$ is a frenemy of $g$ if   $∀x∈ \mathbb{R}, ∃y∈ \mathbb{R}, s.t. x&lt; y \text{ AND } |f(x)-g(x)|&lt;|f(y)-g(y)|$         a) Is the claim “ if $f$ and $g$ are any two functions and $f$ is a rival of $g$ then $f$ is a frenemy of $g$” true or false? If it’s true, prove it, if it’s false then show it with a counter example.   Can someone please show me how to solve for the first question a)? For the the two concepts defined, do I have to first find out if they are true or false before finding if the claims are correct? Is it technically saying that there’s a real number smaller than all other real numbers(y)?
Extremely difficult logic question using conditional statement In this problem, we will only consider (real-valued) functions with domain $\mathbb{R}$. We define two new concepts. Let $f$, and $g$ be two functions.  We say $f$ is a rival of $g$ if   &emsp;&emsp;&emsp;&emsp; $\exists x\in \mathbb{R} \ \text{s.t.} \ \forall y\in \mathbb{R}, x \lt y \implies \lvert f(x)- g(x)\rvert\lt\lvert f(y)-g(y)\rvert$    We say $f$ is a frenemy of $g$ if  &emsp;&emsp;&emsp;&emsp; $\forall x\in\mathbb{R} \ \text{s.t.} \ \  x \lt y \ \ \text{AND} \ \ \lvert f(x)-g(x)\rvert \lt \lvert f(y)-g(y)\rvert$  Below are four claims. Which ones are true and which ones are false? If a claim is true prove it. If a claim is false, show it with a counterexample.  a) If $f$ and $g$ are any two functions and $f$ is a rival of $g$ then $f$ is a frenemy of $g$. b) If $f$ and $g$ are any two functions and $f$ is a frenemy of $g$ then $f$ is a rival of $g$. If $f$ and $g$ are any two functions and $f$ is a rival of $g$, then $g$ is a rival of $f$. Let $f$, $g$, and $h$ be any three functions. If $f$ is a frenemy of $g$ and $g$ is a frenemy of $h$ then $f$ is a frenemy of $h$  What are the four claims stating? Especially for c), the concept is not even present I am having a misunderstanding on how to solve them?
I find in many places mentions of pre-amplifiers being used to boost audio signals prior to feeding to a power amplifier.  Wikipedia's article on  gives some hints at their usage, but it doesn't really explain why it would be necessary to have two amplifiers in series rather than a single one with greater gain.  If a pre-amp is just a power amp with a smaller gain, what's the point, and why the distinction between the two kinds? Is the distinction in function, or simply purpose? How to make a file (e.g. a .sh script) executable, so it can be run from a terminal I have a script.sh file and type of this file is shellscript file. I want to make this file as application/x-executable file. How can I make it? Sturm-Liouville Questions In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! My son is 5 years old and sexually active I am a 20 year old female student and I have a 5 year old son who is really sexually active. For instance, he kisses girls by force, he forces them to lick his penis, he "fingers" them, he speaks about sex openly with everyone except me, his father and my family.  There is a whole lot more that he does and the most frustrating part is that when teachers ask him where he saw those things, he says he learned it from me.  We do not even live together.  Currently I am in Cape Town doing my degree and he is in KZN with my mother.  Since he was 8 months old, he went to live with his father because I had to finish high school.  We never lived together. We have tried talking to him and it hasn't helped.  We've punished him but still he keeps on getting worse. What should I do?  I am really angry, embarrassed and confused. If I am not mistaken, this is his third year acting like this. Answer textbox's content starts overlapping when adding new lines in middle on Android app When typing/editing an answer, sometimes if I scroll the editor textbox, go upwards and add lines, the text will start overlapping each other. The behaviour is that when you add a newline, the text below it does not move downwards.  Minimizing soft keyboard fixes this.  This does not affect the Ask Question box in my test.    I am using Android 4.1.2 Idempotents in a ring without unity (rng) and no zero divisors. Question: Given a ring without unity and with no zero-divisors, is it possible that there are idempotents other than zero?   Def: $a$ is idempotent if $a^2 = a$.   Originally the problem was to show that $1$ and $0$ are the only idempotents in a ring with unity and no zero-divisors, but I wonder what happens if we remove the unity condition.   I am trying to find a ring with idempotents not equal to $0$ or $1$. So far my biggest struggle has been coming up with examples of rings with the given properties.   Does anyone have any hints? How should I attack this problem? Switching motherboard without new install? Get the last item in an array In the 3d-view workspace, we can use a random color to give our separated object a distinct look.  What if I don't like the result and I want to randomize the color again?   Swapping the setting doesn't change the random seed in color.    (Blender 2.80.64.) Why is empty set an open set? I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition. The definition of an open set: a set S in n-dimensional space is called open if all its points are interior points. How to keep leading zeros in PHP integer Currently I am using Outlook 2010, and sometimes customers claim that they will received an attachment in Windows.dat format.  I have searched online for possible solutions, but mostly the suggestion is to change the sending format to plain text. However, by using plain text format, there is some limitation.  So I just wonder, besides the plain text format, is there a ways to solve the winmail.dat issue?  Besides, does plain text format really 100% avoid the issue? Trying to find a story about an unborn telepathic baby who can save a spaceship from doom Why are properties lost in the Cayley–Dickson construction? Motivating question:    I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$  and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering,  commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.   Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).   I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?  * I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties. I am trying to learn data in higher space into lower space. To have a clue, I'd like to know how to transform the data in the image below into a lower dimension preserving the structure. Hope to hear some explanations and what should I study to learn mapping data to lower dimension? Does a Nikon DX lens on a FX camera get subjected to the 1.5X crop factor? Close and flag links do nothing on mobile site I am using Blender to mix the video from my camera with the audio from my external recording setup.  It works great mostly and I am very pleased with it.  But the problem is that I can't start the audio recorder an even number of thirtieths of a second from when I started the camera.  And Blender seems to only let me move the audio track in the VSE by whole frames (1/30 of a second).  Here are a couple screenshots:   The track on top is the external audio and the one on the bottom is the camera's audio.  Between the two screenshots I have only moved the audio track by one frame.  But as you can see, I need to move it by about 1/2 frame to line up the audio properly with the video.  So can I move a track in the VSE by fractions of a frame? How to Avoid Precision Errors While Creating Numpy Arrays? If I have an adjacency matrix for a graph, can I do a series of matrix operations on the adjacency matrix to find the connected components of the graph?