# Specifying and simplifying sets

I was doing the following question:

Find a simpler way to specify the set $A=\big\{ X: X\in \Bbb N \cap \{-5,3,7,11\} \cup \{-5, -3, 13\} \big\}$

Am I right to suggest a simpler way of specifying the following sets is

$$A=\big\{X:X\in \Bbb N \cup \{-5,-3\} \big\} ~?$$

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Maybe I am wrong, but $\mathbb{N}\cap\{-5,3,7,11\}\cup\{-5,-3,13\}$ seems to be ambiguous. Are there no brackets left? – Julius Apr 15 '12 at 9:06
@julius Yes, sorry it should be {X:X N∩{-5,3,7,11}} U{-5,-3,13} – Xabi Apr 15 '12 at 9:09
Julius is right, the intersection with $\mathbb N$ removes $-5$ from the first finite set but this element is added back, from the union with the second finite set. So you just take the union of the 2 finite sets, disregarding "$\mathbb N\cap$". – plm Apr 15 '12 at 10:59

I'd start by noting that ${\bf N}\cap\{{-5,3,7,11\}}=\{{3,7,11\}}$.
It seems to me that the simplest specification for the set won't have $\bf N$ in it at all. I note that the question has been edited since I posted my answer, and I think the edit has changed the problem. – Gerry Myerson Apr 15 '12 at 9:12
@Fatz: No, you would have $\{x:x\in\{-5,-3,7,11,13\}\}=\{-5,-3,7,11,13\}$ – Dennis Gulko Apr 15 '12 at 9:12