Prove that for a subset $A$ of $\mathbb{R}$, if $|A|=\omega$ then $|\mathbb{R}-A|=2^{\aleph_0}$.  Clearly, $|\mathbb{R}-A|\le 2^{\aleph_0}=|\mathbb{R}|$. It suffices to show that for a countable $A\subseteq \{0,1\}^{\mathbb{N}}$ we have $\{0,1\}^{\mathbb{N}}\le \{0,1\}^{\mathbb{N}}-A$, since $\mathbb{R} \sim \{0,1\}^{\mathbb{N}}.$ The idea: Somehow identify $\mathbb{N}$ in $\{0,1\}^{\mathbb{N}}$. Since $|A|\le \omega$, we can then identify $A$ in $\mathbb{N}$ (all within $\{0,1\}^{\mathbb{N}}$). Thus, we will be able to 'transfer' the elements of $A$ to new elements of $\mathbb{N}$ (in $\{0,1\}^{\mathbb{N}}$). An element not in $A$ is mapped to itself. This way, after we remove $A$, we will still have 'a copy of' $A$ in $\{0,1\}^{\mathbb{N}}-A$, proving that $\{0,1\}^{\mathbb{N}}\le \{0,1\}^{\mathbb{N}}-A$. Is this idea OK? This problem is given before the introduction of AC, and therefore, before the proposition that there is always an injection from $\mathbb{N}$ to an infinite set $X$. This means that, in principle, I can't use the fact to identify $\mathbb{N}$ in $\{0,1\}^{\mathbb{N}}$. Are there more elementary approaches to this problem? Thanks.
I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of Cardinal Arithmetic, which states that for $\kappa$ and $\lambda$ cardinal numbers, the larger infinite and the smaller nonzero, then $\kappa+\lambda=\kappa\cdot\lambda=\max(\kappa,\lambda)$.  I was playing around with the equation $k+\aleph_0=\mathfrak{c}$ for some cardinal $k$. From the above, it follows that $\mathfrak{c}=k+\aleph_0=\max(k,\aleph_0)$, which implies $k=\mathfrak{c}$.  I'm curious, can we still show $k=\mathfrak{c}$ without the Axiom of Choice? Is it maybe possible to bound $\mathfrak{c}-\aleph_0$ above and below by $\mathfrak{c}$? But then I'm not quite sure such algebraic manipulations even mean anything, or work like that here. Certainly normal arithmetic does not! Thanks.
CSS Selector that applies to elements with two classes Rosewill RNX-N600UBE Connectivity issue on Ubuntu 17.04 Reflexive and relation Please give me feedback on my answer to this question. Question:  For all $x,y\in R$ define that $x\equiv y$ if $x^{2}=y^{2}$. Then $\equiv$ is an equivalence relation on $R$, there are infinitely many equivalence classes, one of them consists of one element and the rest consist of two elements. Answer: False, since;  $x\equiv y$   if $x^{2}=y^{2}$  on $R$  To show $\equiv$   is reflexive, we need to show that ,  $\forall x\in R:X\equiv x.$  Let $x\in R$,   $X\equiv x$   if $X^{2}=x^{2}$  , which is obvious.  $[x]\triangleq:\{y\in R/x\equiv y\}$  $[0]=[y\in R/0\equiv y\},$  when $y=0$,  Then $y^{2}=0^{2}=0$  Thus; there are not many infinity many equivalence classes. Variable-name \newcommand with parameters within another \newcommand Hide Command Window of .BAT file that Executes Another .EXE File How can I bind arguments to a Python method to store a nullary functor for later invocation?  Similar to C++'s boost::bind.  For example:  def add(x, y):     return x + y  add_5 = magic_function(add, 5) assert add_5(3) == 8 What features differ between the Xbox 360 and PC versions of Minecraft? What differences does the Xbox 360 version of Minecraft have compared to playing on the PC? To clarify - I'm concerned more about the differences exclusive to each platform, not just differences of features that will probably be added in future updates (for example, crafting). Otherwise, I feel this question could be compared to &quot;What is the difference between Minecraft version x and version y&quot;, which isn't what I'm asking. Can a druid in Ape wildshape use weapons? Why does a candle blow out when we blow on it? Our breath is 16% oxygen and only 4% CO2 Does the Force wane as you age? @Valorum said     ...not to mention the sheer strain that comes from continually using the Force at an advanced age.   To my feeble mind, this doesn't make sense. Approaching 900 years old, Yoda is still as strong as any whippersnapper in terms of Force power and usage. Despite his old age, Obi Wan shouldn't have any problems with using his Force powers.  This got me to thinking, does the Force begin to leave you as you age? Is it a flow of Force power that goes down based on (I hate to say this) a decrease in midichlorian count as you approach your expiration date? How do I get started working with JSON in Apex? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, navigate to the  tag.  This question and answer is owned by the community! If you can improve it and make it more useful, you're welcome to edit it.  I have a specific JSON structure in mind. I need to either deserialize this structure that's coming in from a web service, or I need to serialize data into this structure to transmit it to another system. How should I get started on this project in Apex? Position of "only" in “We 𝒐𝒏𝒍𝒚 have two possibilities” Is there any difference in meaning between the three following sentences?   We only have two possibilities. We have only two possibilities. We have two possibilities only.   I have already read  and I know that the position of only and the emphasis (written or oral) may change the meaning of some sentences, but I am not sure if it does here.  Note: It is clear that “only we have two possibilities” has a different meaning. My question is restricted to the sentences I’ve listed above. Was Yoda wrong about the limits of telekinesis? In Empire Strikes Back, Yoda  that an object's size doesn't matter and that the difficulty of levitating an X-Wing was only in his mind. If the upper limits of telekinesis are determined by belief, what's to stop a Jedi/Sith from pulling a moon out of orbit, moving an entire solar system, or splitting an atom with the Force? On the old style user pages such as mathoverflow I noticed that the long comment on the right is floating lower today than it did previously (as recently as yesterday). I don't know if we care, but I noticed it. See all my questions across all Stack Exchange sites? Stop terminal auto executing when pasting a command Right, this is a slightly annoying feature because something this happens, and other times it doesn't, very odd.  I paste a command into the terminal with a view to tweaking its arguments, and sometimes it executes the second I paste it. Sometimes it doesn't.  This ranges from embarrassing ( posting nonsensical comments on an IRC ) to dangerous.   How can I shut this feature off for good? I never, never, never want the terminal to auto execute when I paste to it. Is there really no way to power a DSLR by USB? Using a function in two different ways Entire function $f$ such that $|f(z)| ≤ K |z|^3$ for $|z|\ge1$ and $f(z) = f(iz)$ for every $z∈C$ Let $f(z)$ be an entire function such that for some constant $K$ , $|f(z)| ≤ K$ $|z|^3&lt;$ for $|z|\ge1$ and $f(z) = f(iz)$ $∀z∈C$ , then which of the following are correct ?  (A)$ f(z) = Kz^3$ $∀ z∈\Bbb{C}$ (B) $f(z)$ is a constant function (C) $f(z)$ is quadratic function (D) No such $f$ exists   My attempt : I can deduce that since $f(z) = f(iz)$ and since $i^4=1 \implies f$ must involve only fourth powers of $z$ . Hence $f(z)=a_1z^4+b_1z^8+c_1z^{16}+ \cdots $-------(1)  Also for $|z| \ge 1 $: let C.   Then $|g(z)| = |f(z)|/|z^3| \le k$ => By the Cauchy estimate theorem we can prove that :  $f(z)/z^3$ is a constant function $= c $   $\implies f(z)=cz^3$ for $|z| &gt; 1$     ...........(2)  From (1) and (2) ; we get  there does not exist a function like this ?  My textbook answer says it's a constant function ? A question in four parts.   What are the main problems which supersymmetry purports to solve?  What would constitute lack of evidence for SUSY at the proposed LHC energy scales (e.g. certain predicted superpartners are not in fact observed)?  Are there alternative theoretical approaches which would address the SUSY problem set and which would still be credible in such an LHC no-SUSY-scenario? Where would LHC-disconfirmation of SUSY leave String Theory?   I would like to think that these four points could be taken together as one question.