# How to express a set of all subsets of another set each missing one member

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 $X'$?

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 $X$.

You could write

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

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

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