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 Feb 18 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? I made a small suggested edit just now. Inform me about it if necessary. Additionally, as comments are hard to read, can you please write back in your answer? Please don't write as comments. Feb 18 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Thanks. Chose yours for bounty for your magnificient effort. What's $S$? Why is $\phi(x) = x \in S \wedge P(x)$ ? And what do you mean 'set $z$ is a function'? Do you mean function as in calculus, a relation $A \to B$ such that each input is related to exactly one output? And how did you rewrite $\{ f(x) : x \in S \}$ as that complicated set underneath it? What does $\in z$ mean? Feb 17 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? @anon Fixed it. Thank you. Additionally, you can edit my posts if there are other problems. Please do. Feb 17 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Voted up. Sorry for the confusing $\yen$. I changed it. Do you understand it now? I don't understand your answer sorry. Can you simplify or write more? I've never taken set theory so at this time, I don't know and won't need to know 'syntax of first-order set-theory', wff (this means well-formed formulas, yes ?), constant symbol, and the like? Feb 17 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? I made a small suggested edit just now. Inform me about it if necessary. Additionally, as there are a lot of subscripts, can you please write back in your answer? Please don't write as comments. Feb 17 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Thank you. Voted up. Can you please simplify your answer? It looks very magnificent and prime. I never took set theory. I don't know what are ZF, extensionality, Fraenkel's, many-one, class function, Cantor's arithmetic of infinities, and the like. Feb 3 comment Visualize $A_4$ and $\langle x, z\rangle$ isomorphic to the Klein 4 group thanks. i upvoted for you. can you please reply to my edit that i made in my question --- in your answer? Feb 2 comment Visualize left, right cosets and conjugation thanks. I upvoted for you. can you please reply to my edit in my question? Can you please write in your answer and not in the comment box? Feb 2 comment Animations or Pictures of Group of Rigid Motions (or Rotations) of the Cube thanks for answering. Feb 2 comment Visualize $C_2 \times C_4$ is normal subgroup Can you please elaborate? I know Abelian groups are normal subgroups but I'm asking about visualization??? Feb 2 comment Visualize cosets of $\left<(0,1)\right>$ partition $C_3 \times C_3$ thanks. can you please rewrite your answer without quotient groups? Book hasn't introduced them. Jan 25 comment Visualize cosets of $\left<(0,1)\right>$ partition $C_3 \times C_3$ @DonAntonio I spent the past couple of days going through this. I am only posting at once what I do not understand. I will stew over this more. Jan 10 comment Animations or Pictures of Group of Rigid Motions (or Rotations) of the Cube thanks. i upvoted for you. Nov 23 comment Intuition for Multiple Summation becoming One Summation - Nothing too formal/rigorous please @JamesS.Cook: Thanks a lot! It definitely helps! I'll chew this over a bit longer before asking or accepting. Hope this is ok. I wish I could upvote more than once so will upvote your other fine posts. Oct 5 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? +1 - Thanks. Your good answer made me come up with 4 more question in my question. Can you please get back to me on them? Please answer in your answer or as another another - don't answer as a comment please. Oct 5 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? @RahulNarain: I felt this way from math.stackexchange.com/questions/149966/…. Was there something wrong with thinking this? Sep 18 comment $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? I edited my question overhead. Is it better? Let me know if you can't. Feb 17 comment Question on “Proving $f(x) = 0$ everywhere” Thank you very much Dominic Michaelis! Yes, of course! I stepped away. Feb 17 comment Question on “Proving $f(x) = 0$ everywhere” Thank you very much again. You proved $A = B$ by contradiction, which I understand. But what's the intuition for $A = B$? How did you suspect this to be true? Surely, you must've suspected it proving it? Feb 17 comment Question on “Proving $f(x) = 0$ everywhere” @Dominic.Michaelis: Great! Thanks. My last follow-up is: $\large{\text{Question 5:}}$ How do we know $f(a - d) = f(a + d$? The function $f(x)$ doesn't have to be symmetric about $x = a$?