I'm looking to express an operation on collections of elements from a set. In programming, this would more or less be a variadic function. Is there an equivalent in mathematics?
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2Perhaps a function on the power set of a set? – Michael Burr Mar 20 '16 at 01:29
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That could work, but how would I represent the stop of the collection? So let's just go with a finite version with a maximum of 4 elements in the set. How would I designate a collection of two elements? – Daniel Goldman Mar 20 '16 at 01:30
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Are repeats allows? Does order matter? – Michael Burr Mar 20 '16 at 01:30
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Order matters and repeats are allowed. – Daniel Goldman Mar 20 '16 at 01:31
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There aren't pleasant ways to write this, but the set of interest is $\bigcup_{i=0}^\infty A^i$ where $A^i$ is the Cartesian product of your set $A$ with itself $i$ times. See this question. Each element of this set represents an ordered list of finite length. You're talking about a function on this set. – Michael Burr Mar 20 '16 at 01:33
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(Continuing from comments)
Let $A=\{1,2,3,4\}$. Then:
$A^0=\{\emptyset\}$ (this definition is somewhat nonstandard, but helps for your application)
$A^1=\{1,2,3,4\}$.
$A^2=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)\}$.
The sets continue, but get larger and I don't want to write them.
Each element of $\bigcup_{i=0}^\infty A^i$ is an ordered list of elements (throw all the elements of all the different lengths together into a single collection). You don't need a "stop" because the length is part of the element.
Michael Burr
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This should work just fine, and would extend to uncountable indexes in a reasonable way. I'll go with that. Thanks. – Daniel Goldman Mar 20 '16 at 01:41
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