At any gathering of n people must have the same hair colour.  Step 1, if there is one person at the gathering then everyone at the gathering has the same hair colour.  Step 2,  Assume that k gathering of people must have the same hair colour.  Step 3, Some people are gathered. One is sent out of the room, the remaining k people must have the same hair colour. Now bring that person back and send someone else out. You get another k group of people, all who must have the same hair colour.   Therefore all must have the same hair colour.
I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance.
"need not to be neither atomic nor ordered" Is this double/triple negative grammatically sound?  Ordered operations require that space on disk is allocated sequentially, so spatial ordering corresponds to chronology of operations, but actual disk writes need not to be neither atomic nor ordered.   Explanation:   Words neither and nor emphasize on the fact that both conditions are (independently) allowed to remain unsatisfied, that both are optional. To be more formal, "need not be atomic and need not be ordered". But then again, it becomes a concoction of two negations joined by a conjunction. How to replace subscript symbols Lightning Components Basics: Attributes and Expressions Trail The error :      The Packed field is either not using the lightning input component,   the checked value of the item's Packed__c attribute or the correct   type of input.   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What is the best way to make bold math symbols, in particular Greek letters and \nabla? A certain element which makes functionals positive Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there exist an element $x\in X$ such that $f_n(x) \in (0,\infty)$ for all $n$? How to pick the title of a remote webpage 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! This is a hypothetical question about how a CPU operates. If I purchase two identical CPUs, and use one long term (say one year), will it be identical in speed to the unused CPU? Will the number of clock cycles, latency of requests, etc on the used CPU be less than that of the unused CPU?  A supporting argument may be that mechanical devices degrade over time, While a CPU has no moving parts (other than the external fan), it does have circuits that can be damaged by heat, and voltage spikes. Lets say that after a year of intensive use, the circuits degrade and fewer electrons can pass since the pathway is narrower, etc.   Is this the nature of how a CPU operates, or is it simply working or broken, with no speed degradation in between? I have a High poly mesh and I have decided to make the low poly by merging as it is easiest for me. Would it be possible to change the W Hotkey , so that it merges to the center automatically? It is time consuming pressing W > Merge > Center. I understand that division by zero isn't allowed, but we merely just multiplied $f(x) = 1$ by $\frac{x-1}{x-1}$ to get $f(x) = \frac{x-1}{x-1}$ and $a\cdot 1 = 1\cdot a = a$ so they're the same function but with different domain how is this possible?  Or in other words why don't we simplify $f(x) = \frac{x-1}{x-1}$ to $f(x) = 1$ before plotting the points. Is it just defined this way or is there a particular reason ?  Note: my book says the domain of $f(x) = 1$ is $\mathbb{R}$ and the domain of  $f(x) = \frac{x-1}{x-1}$ is $\mathbb{R}$ except $1$. Show that the determinant of $A$ is equal to the product of its eigenvalues Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$.   So I'm having a tough time figuring this one out. I know that I have to work with the characteristic polynomial of the matrix $\det(A-\lambda I)$. But, when considering an $n \times n$ matrix, I do not know how to work out the proof. Should I just use the determinant formula for any $n \times n$ matrix? I'm guessing not, because that is quite complicated. Any insights would be great. My fish's native habitat is rice fields. My fish's native habitat are rice fields.   Which one is correct? I'm pretty sure it's the first, since 'is' modifies 'habitat,' but it still sounds weird... INT comparison without WHERE statement Stop iCloud automatic download to harddrive How do I stop iCloud from downloading files to my hard drive? I opened up a folder and it immediately started downloading all contents Is there a fast way (without looping over all edges/faces/verts) using ops.mesh or bmesh to get the lists of indices associated with mesh islands as shown below? These islands are part of one mesh but are disjoint from each other.    I know how to select the boundaries by running the operator Select boundary Loop or calling bpy.ops.mesh.region_to_loop() but can't find any API calls to return the vert/edge/face indices associated with the individual islands. After Hoth, why did Vader pursue the Millennium Falcon instead of Luke's X-Wing? In The Empire Strikes Back film, the opening crawl says:     The evil lord Darth Vader, obsessed with finding young Skywalker, has dispatched thousands of remote probes into the far reaches of space....   It has also been established in Return of the Jedi that Force users can sense each other:     Chewie whines nervously. Luke stares at the Huge Super Star Destroyer    that looms ever larger before them.      LUKE: Vader's on that ship.   If Darth Vader is obsessed with finding Luke and he can sense his presence, the obvious thing to do would be to follow Luke's X-Wing after the battle of Hoth rather than the Millennium Falcon. Why didn't he? Ubuntu 16.04 software center doesn't have many apps How to format a USB or external drive? Using Ubuntu 11.10, how do I format an external or USB drive? I don't find an option to 'format' when I right click. In the new version, the drives don't come up on the desktop, but instead come on the Launcher. MacBook Pro 2020 Fan Issues I recently purchased a MacBook Pro 2020 13&quot; (1.4 GHz, 8GB Ram, i5). For some reason, when I open Chrome and watch Twitch my computers fans turn on and go crazy. After opening Activity Monitor, I noticed that Google Chrome Helper (Renderer) uses around 50-80% of my CPU. Also, sometimes when I run code on my MacBook Pro, the fans turn on when the program is compiling/running but slow down a few minutes after terminating the program. Also, when watching Twice on Safari, the fans don't come on. However, around 50-60% of the CPU is still being used. Should I be concerned about the CPU usage (I am nervous because I only recently purchased the laptop)? Is there a proof from the first principle that the Lagrangian L = T - V? Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.