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Let $A$ and $B$ be sets. There seem to be two ways of writing $A \subseteq B$: \begin{equation} \forall x(x \in A \implies x \in B) \end{equation} or \begin{equation} (\forall x \in A) x \in B \end{equation}

I usually see the first one written. I am curious to know if the second version is in proper form, logically. Is it a well-formed formula (wff)? Is there any way to prove these two formulas are logically equivalent?

By the way, I am quite interested in this area of question. For example, I commonly want to know when it is better to write \begin{equation} (\forall x \in A) x \in B \end{equation} or \begin{equation} \forall x \in A (x \in B) \end{equation} is there anything I could read that would teach me the difference, and which is more preferable?

MJD
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EthanAlvaree
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    $(\forall x\in A)P(x)$ is usually (not always) understood to be an abbreviation of $(\forall x)(x\in A\implies P(x))$. Most of the other differences you describe are unimportant variations in punctuation; different authors write them differently, and "which one is more preferable" is a matter of taste. Henning Makholm's answer at the other question is a good survey of the variations. – MJD Jan 01 '15 at 19:25
  • The first is a sentence of ZF, the second is not. – André Nicolas Jan 01 '15 at 19:26
  • @MJD is the equivalence of $(\forall x \in A) P(x)$ and $(\forall x) (x \in A \implies P(x))$ taken as a definition or can it be proved? – EthanAlvaree Jan 01 '15 at 19:35
  • That depends on what the definitions are! But in a formal logical system in which $(\forall x)Q(x)$ makes sense, it is typical to define $(\forall x\in S)P(x)$ to be an abbreviation for $(\forall x)(x\in S \implies P(x))$. – MJD Jan 01 '15 at 19:37
  • A definition, or just a shorthand. It is not something that is proved. @EthanAlvaree – Thomas Andrews Jan 01 '15 at 19:37
  • @AndréNicolas Thank you. So the first is a correct formulation in ZFC, that is good to know. So would you agree this implies it is the more "formal" construction of the two? – EthanAlvaree Jan 01 '15 at 19:37
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    "More formal" is meaningless. You will very rarely be using truly formal language, and there's no reason a computer program couldn't just as well learn to read $\foral (x\in S)$. – Thomas Andrews Jan 01 '15 at 19:39
  • @MJD and Thomas Andrews: Got it, thanks! Also, if there are any good books or other readings that discuss these details in the language of logic, please let me know! – EthanAlvaree Jan 01 '15 at 19:41
  • Any introductory book on mathematical logic should discuss this. Many people like the Enderton book. – MJD Jan 01 '15 at 19:42
  • It is the more formal in the sense that the first is formal while the second is not. The trouble with formal expressions is that they can rapidly become hard to read, though this one is fine. There are two ways out, formal abbreviations and informal ones. One also should not underestimate the power of ordinary mathematical English (or French, or Chinese). – André Nicolas Jan 01 '15 at 19:45
  • The second could be formal in a system other than ZFC. ZFC is very widely used, but its restricted quantifiers are an unusual feature that is by no means universal among formal logical systems. I believe you know this, and I am puzzled as to why you would say otherwise. – MJD Jan 01 '15 at 19:56

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