# Definition of a Set in terms of Other Sets

I have a set $S(k)$ well defined only if $k>0$, and another set $X$ well defined only if a condition $cond$ is satisfied.

I want to define $S(k)$ as the union of $S(k-1)$ and $X$ provided that both are well defined. How can I express this ?

If I write $S(k) = S(k-1) \cup X$, this is not correct if $k=1$ or $cond$ is false.

Thank you.

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I think

$S(k) = S(k-1) \cup X$ provided $S(k-1)$ and $X$ are well defined

is perfectly clear, but if you really want notation then how about

$$S(k) = \begin{cases} S(k-1) \cup X & \text{if } k>0 \text{ and } cond = \text{true} \\ S(0) & \text{if } k = 0 \\ \emptyset & \text{otherwise,} \end{cases}$$

or whatever other "otherwise" result you want.

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Thank you. But is it possible to have a formula without the (if else) ? – user3749 May 2 '12 at 16:44
@user3749 I doubt you're going to get much better than that, considering (if else) is a crucial part of your definition of $S(k)$. The "cases" notation above is very common. – Antonio Vargas May 2 '12 at 16:52