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could anyone please let me know the correct reading(sentence form) of set builder notation, confused with different interpretation in different resources.

Many Thanks

Asaf Karagila
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nish1013
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1 Answers1

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The set $\{x\mid\varphi(x)\}$ is "The set of all $x$ such that $\varphi(x)$ holds." Note that sometimes such collection is not a set (e.g. the collection of all sets); and sometimes we wish to limit the elements to be taken from a certain set $A$.

The set $\{x\in A\mid\varphi(x)\}$ is "The set of all $x$ in $A$ such that $\varphi(x)$ is true."

Asaf Karagila
  • 393,674
  • Many thanks for your reply.

    could you please clarify this "Note that sometimes such collection is not a set (e.g. the collection of all sets);" with some examples

     – nish1013 Nov 09 '12 at 14:28
  • @nish1013: There are several paradoxes of naive set theory which show that certain collections cannot be sets. For example the collection of all sets (as I remarked in my answer). – Asaf Karagila Nov 09 '12 at 14:29