Mathematical logic - Adequate sets of connectives. How can I prove that $$\{\sim,\leftrightarrow \}$$ is not an adequate set of connectives?  Please help me with this exercise, I cannot prove it.
Prove that a set of connectives is inadequate It is relatively easy to prove that a given set of connectives is adequate. It suffices to show that the standard connectives can be built from the given set. It is proven that the set $\{\lor, \land, \neg\}$ is adequate, and from that set it can be inferred (applying De Morgan laws and such) that $\{\lor, \neg\}$, $\{\land, \neg\}$ and $\{\to, \neg\}$ are also adequate.  Nevertheless, I'm stuck trying to understand how to prove that a given set of connectives is inadequate. I know I have to prove that a standard connective can't be build using only the connectives of the given set, but I can't figure out how to do it.  FYI, I'm trying to prove that $\{\lor, \land\}$ and $\{\leftrightarrow, \neg\}$ are inadequate sets of connectives.  Thanks in advance.
When I capture packets using the packet analyser wireshark while visiting  I can see packets that use the HTTP protocol (e.g.a GET request). My Get request uses 4 protocols: Ethernet II, IPv4, TCP and HTTP.  For the IPv4 I can see my own local IPv4 address of my pc as the source, for the destination I see the IPv4 address 91.198.174.225. When I go to this address in my browser I get to some wikipedia related website that says'wiki does not exist'. Why am I not seeing the Hydrogen article when I go to that ip address? How do I install the Firefox Developer Edition? If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite.  so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this point, and also $f(x)$ must be continuous at x as well  so that mean that:  $\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h} =  \lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$  and that: $\lim_{x\to x_0^+}=\lim_{x\to x_0^-}=f(x_0)$  I know I should probably use arithmetic limit laws to prove this but I can't see how what I figured out could help me. any help with that?  Thanks! I have seen some very weird for loops when reading other people's code. I have been trying to search for a full syntax explanation for the for loop in C but it is very hard because the word "for" appears in unrelated sentences making the search almost impossible to Google effectively.  This question came to my mind after reading  which made me curious again.  The for here:   for(p=0;p+=(a&amp;1)*b,a!=1;a&gt;&gt;=1,b&lt;&lt;=1);   In the middle condition there is a comma separating the two pieces of code, what does this comma do? The comma on the right side I understand as it makes both a&gt;&gt;=1 and b&lt;&lt;=1.   But within a loop exit condition, what happens? Does it exit when p==0, when a==1 or when both happen?  It would be great if anyone could help me understand this and maybe point me in the direction of a full for loop syntax description. I'm looking into a restful design and would like to use the HTTP methods (POST, GET, ...) and HTTP headers as much as possible. I already found out that the HTTP methods PUT and DELETE are not supported from the browser.  Now I'm looking to get different representations of the same resource and would like to do this by changing the Accept header of the request. Depending on this Accept header, the server can serve a different view on the same resource.  Problem is that I didn't find a way to tell my browser to change this header.   The &lt;a..&gt; tag has a , that can have a mime type, looked like a good candidate but the header was still the browser default (in Firefox it can be changed in about:config with the network.http.accept.default key). Find a parabola touches the line $y=x$ and $y=-x$ at $A(3,3)$ and $B(1,-1)$ If a parabola touches the line $y=x$ and $y=-x$ at $A(3,3)$ and $B(1,-1)$, then find the focus, axis of the parabola and its directrix.   What I thought: Since the 2 tangents are perpendicular,the origin must lie on the directrix and the line joining A and B is a focal chord. Don't know how to proceed from here.. What does a star-preceded property mean in CSS? Assume that $X$ and $Y$ are independent random variables with $X \sim \exp(\lambda)$ and $Y \sim \exp(\mu)$. It is impossible to obtain direct observations of $X$ and $Y$. Instead, we observe the random variables $Z$ and $W$, where $Z = \min\{X,Y\}$ and $W =1$ if $Z=X$ and $W=0$ if $Z=Y$.  How to find the joint distribution of $Z$ and $W$? How much will a poor first semester affect my grad school chances? Is every sufficiently large positive integer of the form $ab + ac + bc + 1$? Is every sufficiently large positive integer $A$ of the form $ab + ac + bc + 1$   where $a,b,c$ are some positive integers larger than some given positive integer $d$ ?  How large is sufficiently large , in other words how do we know our number $A$ is sufficiently large ( as a function of $d$ ) ?    Update :  I read the following about positive integers that are not of the form $ ab + ac + bc $:    And I quote about that sequence above : " Note that for n in this sequence, n+1 is either a prime, twice a prime, the square of a prime, 8 or 16 "  This is why I considered the weird +1 in $ab + ac + bc + 1$  So composites larger than 16 not a prime , twice a prime or the square of a prime are of the form $ab + ac + bc + 1$.  And I wonder why !  I guess that relates strongly to the Original posted problem. Maybe I should have mentioned that before , sorry.  Will Jagy's answer gave the same OEIS link which I already knew. This also explains some of the comments.  Im no expert in genus theory or Galois theory but I assume this has a simple answer. Special Relativity - Regarding the Simultaneity of Events During the Train Paradox What is the physical meaning of $\epsilon_0$? What are the system requirements for each flavour of Ubuntu Desktop? I'm thinking about installing Ubuntu Desktop, but I don't know what flavor is the best for my system. What are the minimum and recommended hardware requirements? What kind of CPU? How much memory? Should I have Hardware Acceleration? What flavor should I use?  This is an attempt of a canonical answer. My answers have the &quot;official  requirements&quot;, the recommended are a mix of official sources and opinion based (along with the answer it's told the source). You can comment or edit if you feel that the information is obsolete or incomplete. It is a good rule of thumb that any system capable of running Windows Vista, 7, 8, x86 (Intel) OS X will almost always be a lot faster with any Ubuntu flavor even if they are lower-spec than described below. How do "Dummy Variables" work? I do not understand how dummy variables work in math.   Suppose we have:  $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$  How is this equivalent to:  $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$  How does this dummy variable system work?   Since $y$ is the dependent variable for $I_1$ How can $y$ itself be and independent variable for $I_2$  ??  Thanks! Is a constant condition more costly than switching shaders? In general, branching in shaders is not a good idea. But now I have a shader with a condition that is constant with respect to the entire draw call. So the branch that is executed is always the same for one draw call.  Is such kind of branching still more costly than having multiple shaders without these branches and switch between them? Evolutionary chronology - if any, which species have evolved in the last 100 years I heared somewhere, I cannot remember where, that there has not been an evolution observed in our lifetime. In life science class, right at the start, we was doing the usual MRSGREN (how to tell if something is a living organism), and the tutor said that you can also tell something is a living organism if it evolves. So I asked if there has been an observed evolution in our lifetime. She said that there had been many but could not recite any off the top of her head, so the next week she handed me a piece of paper with 8 or so websites that aparrently corroborated her claim. I have not got the paper anymore but I recall there been on there the peppered moth or something. So I set out to look for a chronological list of evolution but could not find an authoritative one. I could only find something on wikipedia that indicated that species have been dying out for the past thousand years, but nothing saying that any have evolved. I'm searching how to remove the chapter+section number on the theorem numbering with a section counter.  With no section counter it's like:  \newtheorem{thm}{Title} Chapter1 ----Section 1 --------thm 1 --------thm 2 ----Section 2 --------thm 3 --------thm 4 Chapter2 ----Section 1 --------thm 5 --------thm 6 ----Section 2 --------thm 7 --------thm 8   And with the section counter:  \newtheorem{thm}{Title}[section] Chapter 1 ----Section 1 --------thm 1.1.1 --------thm 1.1.2 ----Section 2 --------thm 1.2.1 --------thm 1.2.2 Chapter 2 ----Section 1 --------thm 2.1.1 --------thm 2.1.2 ----Section 2 --------thm 2.2.1 --------thm 2.2.2   And I want numbering my theorems like this:  \newtheorem???????? Chapter 1 ----Section 1 --------thm 1 --------thm 2 ----Section 2 --------thm 1 --------thm 2 Chapter 2 ----Section 1 --------thm 1 --------thm 2 ----Section 2 --------thm 1 --------thm 2   I know it's always a pain to remove such things, thanks for whoever will help me ^^ if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M  G.M  inequality as follows:  $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get  $$a^3+b^3+abc \ge 4abc-c^3=c(4ab-c^2)$$ so  $$\frac{1}{a^3+b^3+abc} \le \frac{1}{c(4ab-c^2)}$$ But since $a,b,c$ are positive reals  $$4abc-c^3 &lt; 4abc$$ so  $$\frac{1}{4abc-c^3} \gt \frac{1}{4abc}$$  but i am unable to proceed here Listing only directories in UNIX I am able to backup my app's data from the root/data/data/you.package.name directory by just moving it to SD card manually. however, when i reverse the process,                                                                                                               AKA paste the data back to simulate a restore, the app just force closes, i have tried it with a bunch of apps and noticed the same reaction. Where am i wrong?