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Suppose $X=\{x_1, x_2, x_3, \ldots\}$. The set I want is $X'=\{\{x_2, x_3, x_4, \ldots\}, \{x_1, x_3, x_4, \ldots\}, \{x_1, x_2, x_4, \ldots\}, \ldots \}$, but I don't know how to express that set without using ellipses. For my purposes, $X$ is finite but variable.

How can I express $X'$?

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I think you can represent it very concisely using setbuilder notation.

$$X'=\{X\setminus \{a\}\mid a\in X\}$$

Also, this expression is still valid for infinite sets $X$.

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You could write

$$X' = \{A | A \in P(X), |A| = |X|-1 \}$$

where $P(X)$ denotes the powerset of $X$.

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  • $\begingroup$ What if $X$ is an infinite set? $\endgroup$ – Riley Oct 21 '17 at 22:04
  • $\begingroup$ OP asked about finite sets $X$. $\endgroup$ – flawr Oct 21 '17 at 22:06
  • $\begingroup$ Ok, I didn't see that. I thought $X$ was infinite because OP said $X=\{x_1,x_2,x_3,\dots\}$. $\endgroup$ – Riley Oct 21 '17 at 22:10

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