Given the following three predicates: $A(x)$: $x$ is Emily $B(x)$: $x$ is student $C(x,y)$: $x$ is a book of $y$. Translate the following sentence into a predicate logic expression: Emilies' books are books of students. My ideas: I had trouble to translate &quot;Emilies' books&quot;, since knowing that would be easy to construct given sentence. I think using two universal quantifiers would help, but unsure on how to construct &quot;Emilies' books&quot; part.
Assume that given three predicates are presented below: $H(x)$: $x$ is a horse $A(x)$: $x$ is an animal $T(x,y)$: $x$ is a tail of $y$ Then, translate the following inference into an inference using predicate logic expressions and prove whether inference is valid or not (for instance, using natural deduction): Horses are animals.  Horses' tails are tails of animals. My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. My initial idea was to consider similar sentence such as &quot;w is a tail of a horse&quot; to form required inference, but it was not successful. Would be welcomed to hear your ideas about this task.
A secant inequality for convex functions Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?    It seems to me like they are equal definitions in a way.    Can you give me a counter-example?    Thanks R code to solve for probability of normal distributions? I don't understand which R code I am supposed to be using to figure these problems out. I brief explanation of what the code is doing would be amazing. The problems below are two different applications I'm expected to understand and I don't get either. Thank you!  Problem 1. Let Z~N(0,1) For each of the following, find c such that the equation is true.   a) P(Z&lt;c) = 0.8413  b) P(0&lt;Z&lt;c) = 0.3051  c) P(-c&lt;Z&lt;c) = 0.8664   Problem 2. X~N(3,4) compute  a) P(X&gt;3)  b) P(1&lt;X&lt;8)  c) P(-1.5&lt;X&lt;1) The "" formulas often come in handy, but it's not immediately obvious that they would be true.  \begin{align} \sin(\alpha \pm \beta) &amp;= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &amp;= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{align}  So what I want to know is,    How can I prove that these formulas are correct? More importantly, how can I understand these formulas intuitively?   Ideally, I'm looking for answers that make no reference to Calculus, or to , although such answers are still encouraged, for completeness. What happened to this user's rep, badges, and name? This user is only displaying 56 rep.   But surely (s)he has much more than that.        The user has also never given out any bounties.  Also, badges are missing: In iOS 7, when the phone is in Sleep mode and I receive a call, I don't see the "Decline" button. All I see is a green "Answer..." button that requires me to slide to the right to answer the call. Is there anyway we can change that? I know that there are buttons for "Remind Me Later, etc" but I want the "Decline" button.   It works fine when the phone is not in Sleep mode. Mark my own question as favorite? How to create a local APT repository? I would like to build my own local repository on my LAN, so that machines on the LAN can update and upgrade from it. I want to download the packages and store them on my local server so that I can update, upgrade, install, etc, from it without using the internet. "There are currently N bounties, worth X reputation" dynamic ad and the new sidebar width You've probably seen this sidebar ad on one or more -enabled Stack Exchange sites over the years:    The image is dynamically generated from a non-officially SE app that uses the Stack Exchange API to fetch the bounties from a given site.  As far as I understand, the rendered image automatically scales to the allotted ad space - when the sidebar ads were 220x250, it would render in the sidebar as a 220x250 image while still looking quite oversized in the  meta (as it does right here).  With the new 2016 campaign starting, can we test whether this awesome dynamic ad works with the new sidebar width? How to cross-reference items in description lists? Every now and then I use the description list environment  \begin{description} \item[foo]  foo is good \item[bar]  bar is bad \end{description}   to create lists with non-numerical labels. Unfortunately the description list does not support the label-reference pair as normal latex: i.e. if I write  \begin{description} \item[foo] \label{foo} foo is good .... \end{description} .... For good things (see, for example \ref{foo})...    the reference does not point to the item in the list. Instead it points to the closest environment that supports labeling (usually the subsection). So instead of showing the text foo (in boldface), it shows something like 2.2.1 (the subsection corresponding to where the description environment is sitting).  My question: is there a way or a package which allows references to description list items? My current work-around is to just type the text myself in the spot (since I named it myself, and it is not numbered, the formatting is unlikely to change the label =). But one of the good things about the label/reference pairs in latex is the implementation in pdflatex that allows in PDF hyperlinks between references and the objects referred to, this I cannot replicate myself.   Edit: let me rephrase my question. The primary goal is not to display \ref to something other than what LaTeX thinks the \label is. There are many ways to deal with it, some of which outlined in Michael Underwood's helpful answer below. What I am looking at is whether there exists an implementation of a list environment which is like the built-in description environment in that the displayed "title" for the list item is completely customized, while having support for \labels. (The built-in description environment does not support \labels, whereas the built-in enumerate environment does not support customized "title"s.) The enumitem package adds some customizability to the environments, but as far as I can tell it doesn't support exactly what I want. It is a general concept that horizontal forces can only influence physical quantities along horizontal direction; similarly vertical force can produce acceleration etc. only along vertical direction. This is the same concept that helps us study projectile motion as a combination of two independent motions. If this relation is universally true, how come, then friction depend on a force which normal to it? Why are my photos not crisp? I use a Canon EOS Rebel T3i (600D) with the kit lens. I have experimented a lot with different settings and modes, but I feel like I have wasted many good shots that could have looked amazing, if only they were a little bit sharper.  I use auto-focus on most of my pictures and make sure to choose the right focus point, but all of my photos are still a little bit out of focus. When I zoom in, I can see that the pixels are not very crisp or well-aligned.  I am not sure if this issue is caused by my equipment or skills. I have custom fields in contacts form named checkbox1__c, checkbox11__c, checkbox12__c, checkbox13__c.  When I check checkbox1__c then I want checkbox11__c,  checkbox12__c,  checkbox13__c to be unchecked.  When I unchecked checkbox1__c then I want checkbox11__c, checkbox12__c, checkbox13__c to be checked.  Is there any way to do this by using validation-rules? If not no way to this by validation-rules, then what is the best way to do this? What's the difference between  and ,  and ? What's the difference between &lt;b&gt; and &lt;strong&gt;, &lt;i&gt; and &lt;em&gt; in HTML/XHTML? When should you use each? Divergent or not series? How tipe are this series?  $$\sum_{n=2}^\infty \frac 1 {n^2\ln n}$$ But $$\sum_{n=2}^\infty \frac 1 {n\ln n\ln(\ln n)}$$ I used Condensation Test, but I stack after that. Thanks! Origin of the meaning of joe Why can't two series-connected diodes act as a BJT? Another form of the question is: will joining two diodes with wires (pn-np) make a transistor-equivalent?  I read that they are not equivalent, but why? Is exponential map locally a diffeomorphism w.r.t. base point? Let $M$ be a riemannian manifold and $\exp_p: T_pM \rightarrow M$ the exponential map at $p \in M$.  At each point $p\in M$, $\exp_p$ can be restricted to a neighborhood $V$ of $0\in T_pM$ so that $\exp_p|_V$ is a diffeomorphism. Let $U\subset M$ denote image of $V$ under this map.  Choosing an orthonormal basis $\{E_i\}$ for $T_p M$ and using the isomorphism $E: \mathbb R^n \rightarrow T_pM$, where $E(x^1, \dots, x^n)= x^iE_i$, gives normal coordinates $(U, \phi)$, where  $$ \phi := E^{-1} \circ \exp^{-1}_p: U \rightarrow \mathbb R^n $$  My question is: can we fix some $z\in \mathbb R^n$ (small enough) and take some small neighborhood $\widetilde{U}$ of $p$ so that the function $\widetilde \phi$, defined as $$ \widetilde{\phi}_z (p) := \exp_p ( E(z)) $$ is a diffeomorphism when restricted to  $\widetilde{U}$?  (In the definition of $\widetilde \phi$, the isomorphism $E$ will also vary with $p$.) "Email" or "e-mail"? Which way of writing the word: "Email" or "e-mail" is correct? Both variants seem to be in wide use. If both ones are okay, maybe there is a difference in contexts they have been used (one is more formal than the other)? How to escape quotes in shell?