I am currently studying Fitch-style natural deduction and I am having trouble proving  $p ∨ (p → q)$ without being given a premise.  Below is something I've tried. But it doesn't seem right at all. I'm only using basic intro/elim rules with LEM.    Help is greatly appreciated :)
I'm having a bit of trouble proving this with the propositional calculus rules.  $$p \lor (p\implies q)$$ Would someone mind helping me and showing which rules they've used with an explanation! Thanks!
I have a spreadsheet in Google Docs with two data rows and lots of columns like so:    Is it possible to easily convert these rows into two columns and lots of rows (obviously retaining all information and formulas)? In other words, to something like this:   19.9.2009   967,15 €  28.9.2009   950,78 € 12.11.2009   942,03 € ..           ..   There's probably a neat mathematical term for such a conversion, but I don't remember which (matrix something?). Feel free to suggest a better name for the question.  Edit: To clarify, ultimately I want to get rid of the original (two row) version and continue working on the converted one. TRANSPOSE(), , got me very close but not quite there. I want the raw data in the transposed version, not cells that mirror the original ones. Bread and butter is/are my breakfast? Bread and butter (is or are) my breakfast?  And what's the difference between the above sentence and saying "My breakfast is bread and butter"?  Thanks. Evaluate $6^{433} \pmod {21}$ and a proving question Question 1: Denote $a \mod b$ as $a \% b$, where $a$ and $b$ are some integers  Evaluate 12^32475 % 21    The following is what I tried:    12^32475 % 21 = (12^3 % 21)^10825 % 21 = 6^10825 % 21 = ... = 6^433 % 21   But how to continue ?  Question 2:  I don't understand this: n-1= -1 mod n. Where is this from? And does this line become the next line?    Context: my class is discrete math for computer science. So far for related topics, it covers modular inverse, (extended) GCD algorithm, Fermat's Little Theorem and Chinese Remainder Theorem (CRT). How to easily map c++ enums to strings Why isn't the 'len' function inherited by dictionaries and lists in Python example:  a_list = [1, 2, 3] a_list.len() # doesn't work len(a_list) # works   Python being (very) object oriented, I don't understand why the 'len' function isn't inherited by the object.  Plus I keep trying the wrong solution since it appears as the logical one to me 'bridges' on a PCB copper layer I've come across a design where each pad was connected using 4 'bridges' to the GND copper layar. What stands behind these 'bridges'? Why not make a full copper layer with only solder-mask defining the pads? How to use GROUP BY to concatenate strings in SQL Server? How do I get:  id       Name       Value 1          A          4 1          B          8 2          C          9   to  id          Column 1          A:4, B:8 2          C:9 I want to know how to get the X and Y position of HTML elements such as img and div in JavaScript relative to the browser window. How can a "proper" function have a vertical slope? Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us:   Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because that would mean it's not a function as it fails the vertical line test). This function also has a vertical tangent at $x=0$.   My question is: how can we have a function that also has a vertical tangent? To get a vertical tangent we need 2 vertical points, which means that we are not working with a "proper" function as it has multiple y values mapping to a single $x$. How is it possible for a "proper" function to have a vertical tangent?  As I understand, in the graph I pasted we cannot take the derivative of x=0 because the slope is vertical, hence we cannot see the instantaneous rate of change of x to y as the y value is not a value (or many values, which ever way you want to look at it). How is it possible to have a perfectly vertical slope on a function? In this case I can imagine a very steep curve at 0.... but vertical?!? I can't wrap my mind around it. How can we get a vertical slope on a non vertical function? Help with footer always to bottom Why is "bicycle" pronounced differently from other obviously related words? How can I type special symbols on my Android device like ☺ or ♥ as a regular character? How do I find the privileges I have earned and will earn in Stack Overflow? It used to be off the top bar, now its gone  Where has it moved?  EDIT:  It moved onto the profile page !  Right side same row as your name in profile:    Thank you Long site name hides the diamond in profile page Communities Convert generic List/Enumerable to DataTable? I have few methods that returns different Generic Lists.   Exists in .net any class static method or whatever to convert any list into a datatable? The only thing that i can imagine is use Reflection to do this.  IF i have this:  List&lt;Whatever&gt; whatever = new List&lt;Whatever&gt;();   (This next code doesn't work of course, but i would like to have the possibility of:  DataTable dt = (DataTable) whatever; I lost a couple hundred lines of a Vim plugin I wrote and continue back up every so often because I extracted a tar file containing an older version. I decided not to use source control management when I wrote the plugin, since I only update it periodically.  Why doesn't tar ask before overwriting existing files by default? I'm guessing that there's some rationale behind why the default behavior is to overwrite files silently, since we usually want safety over speed and convenience. For example, we usually use #ifdef NDEBUG, not #ifdef DEBUG; similarly, I would expect tar ask me before overwriting any files, and for it to have some flag which I can use to tell it to overwrite files without asking me. Also, how can make it so that tar does ask me by default, without me having to give it any flags? I'm fine with using tools other than tar, if they support this safer default behavior.  Thank you for your time! Error in Obj-C : Expected identifier or '(' What's the best way to return multiple values from a function? I have a function where I need to do something to a string.  I need the function to return a boolean indicating whether or not the operation succeeded, and I also need to return the modified string.    In C#, I would use an out parameter for the string, but there is no equivalent in Python.  I'm still very new to Python and the only thing I can think of is to return a tuple with the boolean and modified string.  Related question: Traditional search-as-you-type on newer Nautilus versions Until now I was using a "traditional" version of Nautilus: that that comes with Ubuntu 10.10. Now I'm using 3.6.3, included with Ubuntu 13.04 (most likely any recent version is equivalent for the scope of my question, anyway).  One major difference between the two is the search-as-you-type behavior: in classic versions of Nautilus, when you typed an input, the file with the closest alphabetical match would become the selected one, the eligible files/folders being restricted to the current directory.  The new behavior is to perform an optionally-global search instead.  Can one configure Nautilus to provide the previous functionality ? Intersection of open affines can be covered by open sets distinguished in *both*affines Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$.  It's not true in general that $U \cap V$ is affine, so if we want to prove basic results about general schemes (e.g. the stuff in section II.3 of Hartshorne), we need some technical means of understanding what goes on on the intersections of open affines.  I've heard that this can be done by covering $U \cap V$ with open sets which are distinguished in both $U$ and $V$, i.e. sets $W \subseteq U \cap V$ with $W \cong \operatorname{Spec} A_f \cong \operatorname{Spec} B_g$ for some $f \in A$ and $g \in B$.  I'm having trouble actually proving this.    So, questions:   Is this actually true? If so, how are these sets constructed?