Prove that $X=\{(x,y) :x\text{ is rational or }y\text{ is rational}\}$ is path connected.   So for every $(x,y)$ in $X$, I need to find a continuous function $f$ on $[a,b]$ such that $f(a)=x$ and $f(b)=y$.   Trying to think about suitable functions, maybe one which sends everything to nearest rational. But don't think such a function exists.
How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected?    Note that $X$ is a topological space with subspace topology $\tau = \left\{ U \cap X \mid U\text{ is open in }\mathbb{R}^2\right\}$.  I only know two things (with respect to path connectedness):   A topological space $X$ is path connected iff for every $p,q \in X$, there is a continuous function $f: [a,b] \to X$ with $f(a)=p$ and $f(b) = q$.    Continuity $f$ is defined as for every open set in $X$, its preimage is open in $[a,b]$ (with subspace topology induced by $\mathbb{R}$ (with standard topology) )   Suppose $X$ and $Y$ are two topological spaces and $f: X \to Y$ is a continuous surjection. If $X$ is path connected, then $Y$ is path connected.
How to draw Venn diagrams (especially: complements) in LaTeX Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition. Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition. What is the difference between local and .local in ubuntu? Echoing path in ubuntu returns following colon separated paths,  /home/stack/bin:/home/stack/.local/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin   In these paths at some places there is local while at others .local is used what is the difference between both? Show finite complement topology is, in fact, a topology Alright, I think we all agree that what happens with the following code is undefined, depending on what is passed,  void deleteForMe(int* pointer) {      delete[] pointer; }   The pointer could be all sorts of different things, and so performing an unconditional delete[] on it is undefined. However, let's assume that we are indeed passing an array pointer,  int main() {      int* arr = new int[5];      deleteForMe(arr);      return 0; }   My question is, in this case where the pointer is an array, who is it that knows this? I mean, from the language/compiler's point of view, it has no idea whether or not arr is an array pointer versus a pointer to a single int. Heck, it doesn't even know whether arr was dynamically created. Yet, if I do the following instead,  int main() {      int* num = new int(1);      deleteForMe(num);      return 0; }   The OS is smart enough to only delete one int and not go on some type of 'killing spree' by deleting the rest of the memory beyond that point (contrast that with strlen and a non-\0-terminated string -- it will keep going until it hits 0).  So whose job is it to remember these things? Does the OS keep some type of record in the background? (I mean, I realise that I started this post by saying that what happens is undefined, but the fact is, the 'killing spree' scenario doesn't happen, so therefore in the practical world someone is remembering.) Why does the introduced part-title in letter appear twice? Inserting random numbers into a table in MYSQL So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?  When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same speed as the wind, they are basically static relative to each other, so no energy is transferred between them (wind and water particles). And if that water sticks to my skin and wind blows, it'll evaporate thus taking energy from my skin and make me feel cold.  Thing is, I don't think that's really the case but even if it is, if I somehow dry out the wind, will it burn me if it's strong enough? And winds can reach some pretty high velocities (though I must admit I'm not sure if they are comparable to movement of atoms in warm bodies etc...).  So. Bottom line. Can I be burned by wind in some perfect scenario? Creating an index of quality from multiple variables to enable rank ordering I have four numeric variables. All of them are measures of soil quality. Higher the variable, higher the quality. The range for all of them is different:  Var1 from 1 to 10  Var2 from 1000 to 2000  Var3 from 150 to 300  Var4 from 0 to 5  I need to combine four variables into single soil quality score which will successfully rank order.  My idea is very simple. Standardize all four variables, sum them up and whatever you get is the score which should rank-order. Do you see any problem with applying this approach. Is there any other (better) approach that you would recommend?  Thanks  Edit:  Thanks guys. A lot of discussion went into "domain expertise"... Agriculture stuff... Whereas I expected more stats-talk. In terms of technique that I will be using... It will probably be simple z-score summation + logistic regression as an experiment. Because vast majority of samples has poor quality 90% I'm going to combine 3 quality categories into one and basically have binary problem (somequality vs no-quality). I kill two birds with one stone. I increase my sample in terms of event rate and I make a use of experts by getting them to clasify my samples. Expert classified samples will then be used to fit log-reg model to maximize level of concordance / discordance with the experts.... How does that sound to you? Is God either immoral or not omnipotent? The usual (Christian) justification for suffering/evil in the world created by a benevolent God is freedom of the will. However, the more interesting question is not about the source of evil (which the free will may very well be) but why God chose to create the world such as it is. Why not create a world with freedom of the will, but without suffering? Even assuming that God is bound by logic such combination doesn't seem to be logically impossible: creatures certainly can do good of their own free will. There are two alternatives.  1) God is powerful enough for all practical intents and purposes, enough to be called God, but not omnipotent. He created only the best of all possible worlds (Leibniz). This position is perhaps the most defensible philosophically, it also avoids logical problems with , etc., but it does diminish deity's status and invites unpleasant questions like what limits God (logic only, inert matter, outer chaos, other beings), is there someone/thing more powerful, etc.    Interestingly enough, the original Christian theodicy wasn't free will. St.Augustine says poetically "the canvas of creation requires both the black and the white paint". "Requires" here can be interpreted differently,  is to explain the black by the "just punishment" for the original sin, which brings us to Adam and Eve, and back to the free will. But if God is still omnipotent and hence omniscient creating creatures he knew would sin, and then punishing them for it is hardly an improvement.   2) However, Augustine can be read more straightforwardly. Evil is "required" for the perfection of the world just like black paint is required for the perfection of a painting (evil can be interpreted as lack of good rather than something onto itself but that is beside the point). God is immoral, and he is beyond our conception of morality, his notion of perfection is his own. This is much bolder, and similar to Plotinus's position on the existence of the One, the One can not exist because it is "prior to all existence". But it does evoke the unpleasant taste of "ends justify the means". It is one thing to use "God is beyond our logic and comprehension" to get rid of unliftable stones, it is quite another to convince people to worship an immoral perfectionist.  Formally, there is another logical possibility of asserting both morality and omnipotence, but defending the morality of creating a world full of suffering, for a higher purpose perhaps. For example, the suffering only appears bad to us, not to an absolute being in its wisdom. But this simply redefines morality and is only different in phraseology from the Augustine's position. The question is asked in human terms and has to be answered in kind. One is free to argue that God is beyond human logic and/or morality, but it still leaves him immoral as commonly understood.   There does not appear to be a way to square this circle, one has to accept Leibniz's or Augustine's alternative (or both) and deal with the downsides. Option 2) looks more consistent, but it is rarely embraced, at least not openly. Perhaps, for "public relations" reasons it is more attractive to dismiss the question with something like "this is beyond us" or change the subject to the free will.      What are the modern theological positions on the morality/omnipotence in catholisism, protestantism, etc.? Is the question relevant to other religions?   EDIT: I am not sure that it came through but by immoral I did not mean morally deficient or evil. Rather at the God's plane of being morality becomes meaningless, like tastes or smells are meaningless when applied to elementary particles. How do I trust a self signed certificate? Book wherein humans were most dangerous/aggressive of numerous species youtube-dl unable to download audio how to resolve that issue? What determines who gets or becomes an MVP? This hasn't been asked yet, but what determines how or who gets an MVP award in CS:GO? After a match, there's always a little blurb that first states who got MVP first, and then a random fact (Player X got Y knife kills, Player X lasted Y rounds without dying, etc.).  Do these two factors both "factor" into getting an MVP award? Is it a set percentage of enemies you kill? Is there a priority list (ie, planting bomb, defusing, etc.)? I'm trying to wrap my head around this for a while now. It's a simple question for what I think will be a not so simple answer. How would you go about modeling one leg of this stool? I would like to do it without using sculpt mode because I can't get things quite clean enough. How is a Kelvin Pod different than a standard Escape pod? How to list .txt files with specific content? How do I list all .txt files with their full path that have a word 'Ubuntu' in them. Ignore non-text files. When comparing values you should ignore the case i.e. UBUNTU or ubuNtu or UbUnTu are treated as equal to the search phrase given. When clicking &quot;edit&quot; I see:  Your edit will be placed in a queue until it is peer reviewed. We welcome edits that make the post easier to understand and more valuable for readers. Because community members review edits, please try to make the post substantially better than how you found it, for example, by fixing grammar or adding additional resources and hyperlinks.  Like:     Likewise, when I am not logged in, clicking &quot;improve this question&quot; or &quot;improve this answer&quot; shows:  You are not . Your edit will be placed in a queue until it is .  And when I click 'Save', I get:  Thanks for submitting an edit. It is only visible to you until it's been approved by trusted community members  Like:     Who can suggest an edit? Where do suggested edits go? Who can vote on a suggested edit? How do suggested edits get reviewed? Can I earn reputation? What about abuse or bad edits? Sometimes the Community user approves or rejects my edit. Why is that? When can I not suggest an edit? Is there a minimum change requirement for a suggested edit? How do I check my suggested edit history? I disagree with the reasons why my suggested edit was rejected. What can I do? How do I know whether there are edits waiting for review? Someone suggested an edit to my own post, and the reviewers took the wrong action. What can I do? Are tag wikis included in this scheme? What do the &quot;Improve Edit&quot; and &quot;Reject and Edit&quot; buttons do? Why are apparently valid links to rejected edits resulting in 404 errors?   For more information about editing, see &quot;&quot; in the . What counts as an attack? I was looking over the class feature of the 'fiend' patron warlock in 5e. I'm a bit confused on what actions the player needs to take before they can use the 'Hurl Through Hell' ability on an enemy. The PHB states:     Starting at 14th level, when you hit a creature with an attack, you can use this feature to instantly transport the target through the lower planes.   I'm not clear on what an 'attack' means in this context. Does it have to be a physical attack like a sword or crossbow? Do magical attacks like Eldritch Blast that require an attack roll count? What about spells with no attack roll? What should I do if a question has changed since I answered it? The other members vote my answer down because it actually doesn't answer the current question.  This kind of editing sometimes happens with  where the OP keeps adding new questions, making my pre-existing answer now incomplete.  But another pattern is when the OP asks an "XY problem" question, an answer points it out, and he changes the question to react to that.  Those answers now not only aren't answers but can look very tangential to people who don't review the edit history.