It is from discrete mathematics notes.  These are part of the biconditional tautologies.   E29. (∀x)A(x) →B ⇔ (∃x)(A(x)→B)   E30. (∃x)A(x) →B ⇔ (∀x)(A(x)→B)   I don't understand how these formulas are said to be equivalent.  What is the difference in interpretation between (∀x)A(x) →B and (∀x)(A(x)→B) ?  How should I interpret those formulas? and how can I prove it? Thank you!
Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$.
Is $\sin^2(z) + \cos^2(z)=1$ still true for $z \in \Bbb{C}$? Is $$\sin^2(z) + \cos^2(z)=1$$ still true for all $z \in \Bbb{C}$? I've tried rewriting it using complex definitions of $\sin$ and $\cos$, and I don't see why it wouldn't, but the text I'm reading asks this question as if it shouldn't hold? Purely "algebraic" proof of Young's Inequality states that if $a, b \geq 0$, $p, q &gt; 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$).  Back when I was in my first course in real analysis, I was assigned this as homework, but I couldn't figure it out.  I kept trying to manipulate the expressions algebraically, and I couldn't get anywhere.  But every proof that I've seen since uses calculus in some way to prove this.  For example, a common proof is based on this  and integration.  The proof on Wikipedia uses the fact that $\log$ is concave, which I believe requires the analytic definition of the logarithm to prove (correct me if I'm wrong).  Can this be proven using just algebraic manipulations?  I know that that is a somewhat vague question, because "algebraic" is not well-defined, but I'm not sure how to make it more rigorous. But for example, the proof when $p = q = 2$ is something I would consider to be "purely algebraic":  $$0 \leq (a - b)^2 = a^2 + b^2 - 2ab,$$ so $$ab \leq \frac{a^2}{2} + \frac{b^2}{2}.$$ I would appreciate if someone could help me write the mathematical equation for the seasonal ARIMA (2,1,0) x (0,2,2) period 12. I'm a little confused with how to go about this. I would prefer an equation involving $Y_t , e_t, θ$ and $Θ$. Adding new ROLES and USERS through REST in GeoServer I use GeoServer stable version 2.14.1 and get error messages when trying to add new roles and users via REST. I suspect that the GeoServer REST documentation is buggy. When I call the REST interface of my GeoServer instance, I can see that the roles are accessible through the URI http://localhost:8080/geoserver/rest/security/roles. Users are under the URI http://localhost:8080/geoserver/rest/security/usergroup/users. I can query both resources via GET request, but adding new resources via POST or PUT will not work. A POST request to create new roles ends up with status code 405 (Method not allowed). After trying to create new users comes the status code 500 (Internal Server Error). For this, the GeoServer-REST documentation has been referencing the paths  /roles/roles/{role}  for new roles () or  /usergroup/users/  for new users () since several GeoServer versions. But when I use them, this always ends with Status code 404 (Not Found). How can I add new roles and users through REST in GeoServer? The workarround from @implssv in  does not work for me. Ubuntu 14.04.5/16.04 and newer on AMD graphics I have 14.04.5/16.04 or newer and an AMD GPU. I'm not getting the performance I want, and some programs say they need things like OpenCL.  I've heard about fglrx, but if I try to install it on my machine, it breaks and I have no GUI.  Is there anything I can do to get my AMD card working to its full potential? How to use "who" vs. "that" I often get confused when trying to use who vs that.  Some examples that often confuse me:  That     The person that went to the store.      The people that went shopping.      The persons that went shopping.      The group that went shopping.   Who     The person who went to the store.      The people who go shopping.   Please explain when to use either for plural subjects and singular subjects.  Animate and inanimate objects as well. How to create a Customized Ubuntu Server ISO? Do I have to code each case of this Grid full of plots separately? Very strange error suddenly appeared running Biber I am on a Mac with MacTex 2020 and ran into a strange problem with Biber. Working on a document all was fine, I did not change the bibliography (says my git) and only fixing typos (not even adding cites) and suddenly latexmk stopped working after a Biber run, while that looked like  INFO - This is Biber 2.14 INFO - Logfile is 'doc.blg' INFO - Reading 'doc.bcf' INFO - Found 31 citekeys in bib section 0 INFO - Processing section 0 INFO - Globbing data source 'literature.bib' INFO - Globbed data source 'literature.bib' to literature.bib INFO - Looking for bibtex format file 'literature.bib' for section 0 INFO - LaTeX decoding ... INFO - Found BibTeX data source 'literature.bib'   so harmless in my view. Then with --debug I additionally get in the end.  INFO - LaTeX decoding ... warning: overriding existing definition of macro "jun" warning: overriding existing definition of macro "jul" warning: overriding existing definition of macro "sep" warning: overriding existing definition of macro "jan" [...] warning: overriding existing definition of macro "nov" INFO - Found BibTeX data source 'literature.bib' Unicode::UCD: failed to find unicore/version in /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc CODE(0x7fd026aeb5a8) CODE(0x7fd026aebb90) at /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib/Biber/Input/file/bibtex.pm line 1011.   It seems unrelated to the library itself since deleting a few entries (going through deleting one half, then the other), the problem persists. It does not happen for an empty literature, but as soon as a first entry is added (just author/journal/title/year/volume/pages - no utf8-only chars) this error message appears.  what does this last error message mean?  How can I solve this? How can an ordered pair be expressed as a set? The difference between log and ln $$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$  Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the difference of the two? I run Windows XP and the wallpaper I used to have got lost. I have backups of the system itself, so if I knew where to look, I could find a copy of the old wallpaper.  I have some idea of the original filename (searching now), and I recall it saves it as a bitmap, but an exact location would be helpful.  It was a photo/JPEG that used to be on the filesystem, and I don't have backups, so the Firefox, and Internet Explorer wallpaper locations are unlikely. I checked %SYSTEMROOT%\system32 as well - so I'm assuming it's somewhere else. Insert into ... values ( SELECT ... FROM ... ) I am trying to INSERT INTO a table using the input from another table. Although this is entirely feasible for many database engines, I always seem to struggle to remember the correct syntax for the SQL engine of the day (, , , , and ).  Is there a silver-bullet syntax coming from an SQL standard (for example, ) that would allow me to insert the values without worrying about the underlying database? When an account is deleted, don't auto delete its negatively scored posts on the meta site Meta is different. Negatively scored questions are not always bad questions, and sometimes they have merit and draw good answers, especially discussions.  When an account (in the main site) is deleted, either by self deletion or by a moderator, all questions with score of -1 or less are being deleted regardless of their answers, both in the main and meta site.  Can you please change this behavior so that the questions on the per-site meta will survive and won't be auto-deleted? Not able to understand the meaning of question? car moving with a speed of 40 km / h can be stopped by applying brakes after at least 2 m. If the same car is moving with a speed of 80 km / h, what is the minimum stopping distance  Does this mean that after 2 m, You will apply brakes and then it stops. What is the retardation value  with which you are brakes.I am not understand the meaning of the values. Possible Duplicate:          I want to know some example of a continuous function which is continuous at exactly one point. We know that $f(x)=\frac{1}{x}$ is continuous everywhere except at $x=0$. But i think this in reverse manner but i dont get any example. So please help me out! Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$ Is there an analytic function $f$ on $B(0,1)$ the open ball with radius 1 such that $f(1/n)=e^{-n}$ for $n=2,3,4,...$? Is there an analytic function $f$ on $B(0,1)\subset\mathbb{C}$ such that $f(1/n)=e^{-n}$ for $n=2,3,4,...$? I know the following doesn't work:  Let $g(z)=\exp(-1/z)$. Then, $f=g$ on a sequence with a limit point in $B(0,1)$ and so $f=g$ on $B(0,1)$. Since $g$ is not $\mathbb{C}$-differentiable at $0$, neither is $f$ and so such a function cannot exist.  This is not the solution because you cannot use the identity principle with a non analytic function like $\exp(-1/z)$ is not analytic at $z=0$. Any help using the identity principle another way? I have heard that one ought to turn the pedals at 90rpm. However, I do so much slower, especially while standing, and would like to understand the argument behind this number (or any other number). So:  What is the optimal cadence of a cyclist. Why? Does it depend on the terrain? Does it depend if the rider is sitting or standing upright? What are some possible consequences of deviating too much from the proposed number?  Possible duplicate of , but I didn't find there the "whys" except for "it will grind your knees." However, I find riding upright with slow cadance no more demanding than walking up a hill with a backpack. long long implementation in 32 bit machine As per c99 standard, size of long long should be minimum 64 bits. How is this implemented in a 32 bit machine (eg. addition or multiplication of 2 long longs). Also, What is the equivalent of long long in C++.