A = B ⇔ (∀x.x ∈ A ⇔ x ∈ B)
What does "∀x.x" mean? This expression means: saying 2 sets are equal, is equivalent to saying they're the same set, right? Thank you.
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$\begingroup$ Looks to me like it means , $\endgroup$– Lee MosherApr 9, 2014 at 12:36
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$\begingroup$ or a ":" instead (mistyping) $\endgroup$– rlartigaApr 9, 2014 at 12:37
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1 Answer
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It is a separator between $\forall x$ and the formula $x\in A\iff x\in B$. It means that the context of $x$ is fixed to be that of the quantified value throughout the rest of the formula.
answered Apr 9, 2014 at 12:39
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$\begingroup$ Oh, like parentheses (but until the end of the expression)? It makes sense... Not exactly like parentheses but I think I got it. Thank you. $\endgroup$– SadSevenApr 9, 2014 at 12:41
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$\begingroup$ @SadSeven - exactly like parentheses; you may (and I think it is better) rewrite it as : $A = B \leftrightarrow (\forall x)(x \in A \leftrightarrow x \in B)$. $\endgroup$ Apr 9, 2014 at 12:46
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$\begingroup$ @MauroALLEGRANZA Alright, thank you very much! :) $\endgroup$– SadSevenApr 9, 2014 at 12:52