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In the context of submodular functions, I encountered the following statement :

For a vector $x \in \mathbb{R}^V$ and a subset $Y \subseteq V$ we define the expression $x(Y)$ as $\sum_{u \in Y}x(u)$.

$V$ is a set.

What does this statement mean ?

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    Just to nitpick, why is the union disjoint here?2011-06-08
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    @Alexei I am sorry i don't understand what you are saying2011-06-08
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    Oh, I get it now, silly me :)2011-06-08

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For sets $X$ and $Y$ the notation $X^Y$ means the following:

$$ X^Y = \{f:Y \to X \mbox{ function}\} $$

if $X$ is a field, then $X^Y$ can be given a structure of vector space over $X$ with the obvious point-wise operations.

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    To elaborate further, one can see that this is in some way consistent with the notation $\mathbb{R}^n$ which can be viewed as the set of functions from a set of $n$ elements to $\mathbb{R}.$2011-06-08
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    @Jay how can the notation $x \in \mathbb{R}^n$ be interpreted as you suggest ? I only know of the interpretation that x is simply a vector of n elements each one of which belongs to $\mathbb{R}$. Please elaborate a little more.2011-06-08
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    @AnkurVijay an n-tuple is simply a function from $\{1,\ldots,n\}$ (or $n$ as an ordinal) to $\mathbb{R}$.2011-06-08
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    @Alexei i still dont understand how a single value is being assigned to an n tuple.2011-06-08
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    @AnkurVijay to $f:\{1, \dots, n\} \to \mathbb R$ assign $(f(1), \dots, f(n))$2011-06-08
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    @Giacomo sorry for being a little dumb here, but could you explain to me with an example ? $x \in \mathbb{R}^n$ to me means an n tuple each element of which is in $\mathbb{R}$. For example x = (5,3,7.5, -10) is a 4-tuple. Now how do i give this a single value belonging to $\mathbb{R}$. What is $f$ in your comment above ?2011-06-08
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    $4$-tuple $x=(5,3,7.5,-10)$ corresponds to function $f$, with domain $\{1,2,3,4\}$ where $f(1) = 5, f(2)=3, f(3)=7.5, f(4)=-10$2011-06-08
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    @GEdgar thankyou very much, now i understand what is being suggested2011-06-08
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    @AnkurVijay: perhaps this old answer of mine is helpful: http://math.stackexchange.com/a/51062/26142012-02-02
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It refers to functions that go from Y to X.