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In my set theory book (A book of set theory, Charles,C,Pinter) domain and range of graph $G$ is defined by

$dom G=\{x:\exists y \ni (x,y)\in G\}$
$ran G=\{y:\exists x \ni (x,y)\in G\}$

But I don't know what $\exists x \ni (x,y)$ exactly means. Ordered pair is defined with $(x,y)=\{\{x\},\{x,y\}\}$ so $x\in \{x\}\in \{\{x\},\{x,y\}\}=(x,y)$. I think we should write $x \in (x,y)$ instead of $x \ni (x,y)$ What makes that definition possible?

Asaf Karagila
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noobgi
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    The second symbols is "such that". – The Count Mar 18 '17 at 01:51
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    $\exists y \ni (x,y)\in G$ read as "there exists $y$ such that $(x,y)\in G$ " – Juniven Acapulco Mar 18 '17 at 01:52
  • The first one should say: There exists y with the property that (x,y) is a member of G. (Not necessarily in those exact words). Normally I would expect to bracket or a vertical line where the backwards epsilon appears. It may be a typo in the book. Does he give a definition of his use of this symbol? BTW there is another widely-used def;n of (x,y) , namely ${x, {x,y}}$. – DanielWainfleet Mar 18 '17 at 01:56
  • pls see: https://books.google.de/books?id=q1KVAwAAQBAJ&lpg=PA90&hl=de&pg=PA24&output=embed"width=500 height=500 – mle Mar 18 '17 at 02:05
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    @ΘΣΦGenSan First, thanks for what symbol means. Second, is {{x},{x,y}} a set because {x} and {x,y} are set? and x is an element of {x} and {x} is an element of {{x},{x,y}}. Is there something wrong with my idea? – noobgi Mar 18 '17 at 02:07
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    @TheCount Thanks! – noobgi Mar 18 '17 at 02:11
  • Formally, $x\in(x,y)$ is not correct, but that's also not important. You just want some definition of the pair such that from the pair you can conclude what the value of the first coordinate and what the value of the second coordinate is. It is a common exercise in set theory to prove $\forall x,y,z,u ((x,y)=(z,u)\rightarrow (x=z\wedge y=u))$. There is no connection between $\in$ and $\ni$ in meaning. Your reasoning why $(x,y)$ is a set, is correct. If you know ZFC, it uses the Pairing and the Separation Axioms. – martin.koeberl Mar 18 '17 at 02:11
  • @noobgi You're welcome. Another common (more common, to me, but some will differ) notation is "s.t.". – The Count Mar 18 '17 at 02:13
  • @martin.koeberl I agree with x∈(x,y) is not important. But is that mean x is not an element of {{x},{x,y}}? – noobgi Mar 18 '17 at 02:31
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    Oh yes, I was referring to this definition. The axiom of Foundation implies this. – martin.koeberl Mar 18 '17 at 02:32
  • @martin.koeberl Thanks! Now I understood why my idea is wrong by axiom – noobgi Mar 18 '17 at 02:53

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