Is image of recursive set under recursive function recursive? Given $A$ - recursive set and function $f$ which is also recursive. Is $f(A)$ recursive?  I think that it isn't recursive, but how to prove it?
Example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive? Browsing wikipedia, I stumbled on the following: "The image of a computable set under a total computable bijection is computable." Given the form of the theorem, there must be some example of a example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive (otherwise the statement would have presumably been stronger). But I'm having difficulty dreaming up a case where this happens.
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$.  Is there a characterization of the groups in which this property holds?  If this question is too broad, I might ask if such a characterization exists for $p$-groups.    History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{  s.t. } H \cong G/N$, and crossposted this to .  I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property.  After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable. Entropy of sum is sum of entropies Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$. Let $x$ and $y$ be positive integers such that $xy \mid  x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks Is it possible to animate which characters of a text object are assigned to a material? In .NET, which loop runs faster, 'for' or 'foreach'? Escaped line break tag in tooltip on election page What book features people living in a tree who move up it in a bubble? Should I write "X and I", "X and me", "I and X", or "me and X" in a conjoined object? Applied modifiers are not visible Evaluation of $\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}$ Rails: What's a good way to validate links (URLs)? How can you programmatically tell an HTML SELECT to drop down (for example, due to mouseover)? How can you programmatically tell an HTML select to drop down (for example, due to mouseover)? Which stats / gear can get the best our of wasteland exploration? Trying to get the best out of wasteland exploration here! I get it that you'd obviously need a big gun for survival out there, but apart from that, I'm wondering which stats / gear can get the best out of exploration.    How would you choose who to send out exploring from your dwellers. Which stat is the most important to choose explorers? Strength or endurance maybe? Does it make much of a difference equipping combat outfits? If so, whats the most important stat to boost through outfits? Are there integer solutions to $9^x - 8^y = 1$? Windows batch command(s) to read first line from text file How can I read the first line from a text file using a Windows batch file? Since the file is large I only want to deal with the first line. How to get directory size in PHP Density of Solid States of Compounds One of the wonderful properties of water (as my high school biology teacher would say) is that in its solid form, it is lighter than its liquid form. This means that when temperatures drop below 0 degrees Celsius, the top layer of water on, say, a lake freezes first. This works out pretty well for any fish or other aquatic creatures living underneath, because the lower layers of water will not freeze.  I know that this is an exception to the general rule of solid and liquid states: A given substance, when transformed into its solid state, will generally sink in a container of its liquid state. My question is this: What other substances are exceptions to this rule (if any?). What features do they share with water that are responsible for this? Are there Ubuntu security notices feeds for specific releases Do objects rotate around the torque vector or its center? If a sphere has a torque vector coming out of it at point A, would the sphere rotate about its center or the axis of the torque vector? Inheriting from a UserControl in WPF I am fairly new to WPF and I am having a problem with inheriting from a user control.  I created a User Control and now I need to inherit from that control and add some more functionality.  Has anyone does this sort of thing before?  Any help would be greatly appreciated.  Thank you