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 Feb27 awarded Popular Question Jul2 awarded Curious May30 awarded Investor May6 awarded Popular Question Feb18 revised $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Offered space due to the many symbols involved. Hard to read without space. Improved formatting. Feb18 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. Feb18 suggested approved edit on $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Feb18 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? Feb18 revised $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Corrections Feb17 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. Feb17 revised $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? added 139 characters in body Feb17 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? Feb17 revised $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Clarified abbreviation. corrected typo Feb17 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. Feb17 suggested approved edit on $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Feb17 revised $\{x : P(x)\}$ vs. $\{P(x) : x\}$ -— When are these set-builder notations the same and different? Amended $. Joined an equality broken across two lines. Feb17 suggested approved edit on$ \{x : P(x)\} $vs.$ \{P(x) : x\} $-— When are these set-builder notations the same and different? Feb17 comment$ \{x : P(x)\} $vs.$ \{P(x) : x\} $-— When are these set-builder notations the same and different? Thank you. Voted up. Can you please simplify your answer? It looks very magnificent and prime. I never took set theory. I don't know what are ZF, extensionality, Fraenkel's, many-one, class function, Cantor's arithmetic of infinities, and the like. Feb17 revised$ \{x : P(x)\} $vs.$ \{P(x) : x\} \$ -— When are these set-builder notations the same and different? added 245 characters in body Feb3 asked Visualize meaning of quotient in quotient map, group - etc?