Explain the flaw in the following induction argument which shows all of Lucas’ toys are the same colour.  Proof: We will show by induction that: for every integer $n\ge1$, in any group of $n$ of Lucas’ toys, all the toys in this group are the same colour.  Basis: If Lucas had only one toy, then clearly all these toys are one colour, so the result holds for $n=1$.  Induction Hypothesis: Suppose for a fixed but arbitrary integer $k \ge 1$, in any group of $k$ of Lucas’ toys, all the toys in this group are the same colour.  Induction Step: Let $n = k + 1$. Take any fixed but arbitrary group of $k + 1$ of Lucas’ toys. Pick out an arbitrary toy (Toy A) from this group. Then all the remaining $k$ toys in the group are the same colour, by the induction hypothesis. Adding this toy back into the group and removing another toy (Toy B), we get a group of $k$ toys which contain Toy A. By the induction hypothesis, these $k$ toys are all the same colour. Thus all the toys in the group must be the same colour. Since the group chosen was arbitrary, in any group of $k + 1$ of Lucas’ toys, all the toys in this group are the same colour.  Conclusion: By the PMI, for all integers $n \ge 1$ any group of $n$ of Lucas’ toys are all the same colour. Therefore, since Lucas has a finite number of toys, all of Lucas’ toys are the same 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.
How can I avoid soft dull "spots" in my pictures? Recently, I've been doing a lot more photography, particularly with my 100-400mm telephoto and 16-35mm wide. I am skilled enough that I can capture the shot mostly as I want with just the camera gear, requiring minimal post processing. I'm a huge fan of ETTR now, so most of my shots are overexposed to support more shadow DR and improve the noise performance of my el-chepo 450D body.  I've noticed quite a bit lately during post processing in lightroom that my sky's or any other lighter areas of my shots tend have spots on them. They are soft and dull to one degree or another (seems to depend on the zoom factor), so probably something either on the sensor or the lens. I'm having a hard time figuring out exactly what they are, as I try to use my lens pen brush before any shoot, yet I still have them. I am having to make quite a few healing corrections post-process to get rid of them.  Is there an easy way to identify where the spots are on my lenses? Once identified, if a quick brush of the outer lens element with a lens pen (or soft camel hair brush) doesn't do the trick, does that mean the spots are on my sensor? Is it possible there is particulate inside the lens body (all of my gear is less than a year old outside of my 450D body, and its all Canon L-Series gear, so I would home that I don't already have detrimental particulates floating around inside my lenses messing up my shots. :|)  Here are a couple examples. The spots are harder to see on screen, and I first noticed them when I printed a few of my more recent shots. There are quite a few, and the shot the examples below were taken from, where the sky took up about 30% of the top edge of the photo, there were probably 10-12 different spots, some of them darker, many of them lighter. The darker ones are the ones that are very visible in prints, where as the softer ones are not as visible, but still apparent enough to ruin an expensive print. How can I detect if my apple device supports Bluetooth Low Energy What is the difference between the "...", '...', $'...', and $"..." quotes in the shell? Sometimes I see shell scripts use all of these different ways of quoting some text: &quot;...&quot;, '...', $'...', and $&quot;...&quot;. Why are there so many different kinds of quote being used? Do they behave differently or affect what I can do inside them? Loopy integral $\int xe^x\sin(x)$ After logging in on terminal with user name and password, how do I start the GUI? Using Regex to generate Strings rather than match them "Leave a comment" link even when you can't On  I disabled  comments for "USDCAD options vs CADUSD options arbitrage?", but it  still has a "Leave a Comment" link.   Clicking the link takes you to    where it, of course, shows that you can't leave a comment.   The timeline page correctly does not show the 'comment' link:    How to fix? I have been studying the pronunciation of Greek-derived words in English, and I've found an odd anomaly. There are (at least) two possible pronunciation patterns for plural word-forms that end in -&#8288;mata (which correspond to singular word-forms that end in -&#8288;ma).   Antepenult stress (expected)  It appears that it is always possible to place the stress on the antepenult (third-to-last) syllable, such as:    /ˈlɛmə/, /ˈlɛmətə/  /ˈskiːmə/, /ˈskiːmətə/  /ˌtɛrəˈtoʊmə/, /ˌtɛrəˈtoʊmətə/~/ˌtɛrəˈtɒmətə/  /ˈstɪɡmə/, /ˈstɪɡmətə/   This is what would be expected from  (I believe most or all of these words came through Latin before entering English), since the penultimate "a" in the original Greek words is short.  Maybe Latin vowel quality and stress rules are not very relevant; certainly, few English speakers know about them nowadays. And even with antepenult stress, some of these pronunciations do not seem to follow all of the traditional rules for pronouncing Latin-derived terms in English. I was reminded by some comments in this  that  in antepenult stressed syllables, we would expect tense ("long") vowels to be shortened due to  (TSL), yielding pronunciations like /ˌtɛrəˈtɒmətə/ (which I just noticed is in fact listed in the Oxford English dictionary) and ?/ˈskɛmətə/ (which is not). (Compare to genus~genera /dʒiːnəs/~/dʒɛnərə/, and perhaps opus~opera /oʊpəs/~/ɒpərə/). However, there are other Latinate plural forms that typically don't have trisyllabic laxing (matrices, also I think helices), or where both the lax- and tense-voweled pronunciations coexist (codices, apices), so I guess the apparent lack of it in schemata should not be surprising. (There are even some exceptions to TSL in singular nouns like obesity.)  Penult stress (unexpected)  The other pronunciation seems to exist only for a couple of words, in which the penultimate "a" is commonly stressed and broadened: stigma ~ stigmáta  (listed by both the Merriam-Webster dictionary and the ), and also sometimes schéma ~ schemáta /skiːˈmɑːtə/ (listed only by the ). (Neither the Merriam-Webster dictionary nor the Oxford Learner's Dictionaries list *"lemmáta").  I'm wondering why this pronunciation exists. It's an irregular plural either way, and putting the stress on the penultimate syllable creates a stress change between the singular and the plural forms.  (It's true that all of these words are rare, and some people might pronounce some of their plurals differently. The pronunciations I give here are simply the ones listed in dictionaries.)  The plural form "stigmata" is probably most common in religious or theological contexts (where it refers to the wounds of Christ), so at first I wondered if the penultimate stress was supposed to be closer to the original Greek pronunciation, but according to , the stress is on the antepenult in Greek as well.  So, what's up with this? Did the stress perhaps change in Latin over time, is this a tradition of Greek accentuation that I'm not familiar with, is this a simple matter of ignorance of the original position of the stress that has now become entrenched with time and usage, or is this somehow explainable by established English stress patterns? (For example, do English speakers have some natural preference for a penultimate stress on polysyllabic words?)  If nobody knows of a reason, I suppose I can just put it down to people encountering the word mainly in text, not knowing which syllable to stress, and choosing the penult because it sounds better to them.  (I asked this question a while back on , but nobody knew of any reason.) What is the difference between a URI, a URL and a URN? People talk about URLs, URIs, and URNs as if they're different things, but they look the same to the naked eye.  What are the distinguishable differences between them? Hi Blender coders and artists.  I am updating one of my add-ons for Blender 2.8 and have run into some  issues finding the appropriate way to hide a collection that is added to my scene.  I want to deselect the eyeball on the toolbar with a python command in a script but I don't want to disable the collection in the view-port entirely as it is used as a collection in a particle system.   I just want to deselect the eyeball circled in green below.  Anyone know the python API for this? Computing limit of $(1+1/n)^{n^2}$ Other word for person who does something to others but does'nt like it done to them This is a duplicate question but has still not been answered for me. I'm so close to the exact word - I know there is one out there - but the closest word I can find is a hypocrite --     a person who claims to have certain moral principles or beliefs but behaves in a way that shows they are not sincere.    The word I'm looking for is a person who does not claim things, but does thing to another and they consider it wrong or immoral to have done to them. Manual/automatic line breaks and text alignment in TikZ nodes I downloaded Ubuntu 14.04 (64 bit) lts. I made a bootable usb stick. But I keep receiving this error message upon booting:  SYSLINUX 4.04 EDD 20110518 Copyright (C) 1994-2011 H. Peter Anvin et al Unknown keyword in configuration file: gfxboot.c32: not a COM32R image How do I install Ubuntu Server (step-by-step)? After removing any part the rest can be split evenly. Consequences? Let $S$ be a finite collection of real numbers (not necessarily distinct). If any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same sum ; then is it true that all elements of $S$ are equal ?  (I know the result is true if the &quot;reals&quot; are replaced by &quot;integers&quot;.) How to increase battery life on Ubuntu 20.04 and what power saving software should I install, if any? I am typing to get a sale amount (by input) to be multiplied by a defined sales tax (0.08) and then have it print the total amount (sales tax times sale amount).  I run into this error. Anyone know what could be wrong or have any suggestions?   salesAmount = raw_input (["Insert sale amount here \n"]) ['Insert sale amount here \n']20.99 &gt;&gt;&gt; salesTax = 0.08 &gt;&gt;&gt; totalAmount = salesAmount * salesTax  Traceback (most recent call last):   File "&lt;pyshell#57&gt;", line 1, in &lt;module&gt;     totalAmount = salesAmount * salesTax TypeError: can't multiply sequence by non-int of type 'float' I have a list of cards for a deck I want to build or buy. What should I do to check whether that deck is legal in a particular format (Standard, for example)? Table with 100 % width and vertical / horizontal alignment I want to draw the following table in LaTeX. In the first column the text should be left aligned. The text in all other cells should be centered. The first problem occurs with D in the first row. If I write a \centering in front of D, I get a LaTeX error. The second problem is, that I want to vertically align the text in all cells but I don't know how. I have found a solution but it doesn't work if the width of the first column is specified.  \documentclass{article} \usepackage{blindtext} \usepackage{tabularx}  \begin{document}   \begin{tabularx}{\textwidth}{|p{4cm}|X|X|X|X|}     \hline                &amp; \centering A &amp; \centering B &amp; \centering C &amp; D\\     \hline     \blindtext &amp; 123 &amp; 123 &amp; 123 &amp; 123\\     \hline   \end{tabularx} \end{document}