Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
Just to nitpick, why is the union disjoint here? – Alexei Averchenko Jun 8 '11 at 14:44
@Alexei I am sorry i don't understand what you are saying – AnkurVijay Jun 8 '11 at 15:25
Oh, I get it now, silly me :) – Alexei Averchenko Jun 8 '11 at 19:47
up vote 6 down vote accepted

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.

share|cite|improve this answer
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 $\mathbb{R}.$ – Kopper Jun 8 '11 at 7:11
@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. – AnkurVijay Jun 8 '11 at 7:30
@AnkurVijay an n-tuple is simply a function from $\{1,\ldots,n\}$ (or $n$ as an ordinal) to $\mathbb{R}$. – Alexei Averchenko Jun 8 '11 at 7:42
@Alexei i still dont understand how a single value is being assigned to an n tuple. – AnkurVijay Jun 8 '11 at 7:44
$4$-tuple $x=(5,3,7.5,-10)$ corresponds to function $f$, with domain $\{1,2,3,4\}$ where $f(1) = 5, f(2)=3, f(3)=7.5, f(4)=-10$ – GEdgar Jun 8 '11 at 13:19

It refers to functions that go from Y to X.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.