# How to understand set as exponential?

• May 1st 2012, 06:54 AM
colruyt
How to understand set as exponential?
I came across a paper, where the notation is:
$\displaystyle 2^\Theta$
where $\displaystyle \Theta$ is a set,
Normally we see exponential such as $\displaystyle a^n$, where n is a real number.
How should I interpret 2^Theta, is this a matrix where in each entry the base is 2?

Thanks for help.
• May 1st 2012, 07:06 AM
ModusPonens
Re: How to understand set as exponential?
It's the set of all functions from theta to {0,1}.
• May 1st 2012, 07:06 AM
Plato
Re: How to understand set as exponential?
Quote:

Originally Posted by colruyt
I came across a paper, where the notation is:
$\displaystyle 2^\Theta$
where $\displaystyle \Theta$ is a set,
Normally we see exponential such as $\displaystyle a^n$, where n is a real number.
How should I interpret 2^Theta, is this a matrix where in each entry the base is 2?

There are two principal meaning the symbol $\displaystyle 2^A$ where $\displaystyle A$ is set.
It stands for the set of all functions from $\displaystyle A$ to $\displaystyle \{0,1\}.$
Some authors use it to be short hand for the power set, the set of all subsets of $\displaystyle A$.
• May 1st 2012, 07:22 AM
colruyt
Re: How to understand set as exponential?
Thanks a lot.
I don't understhand the first meaning:
Quote:

It stands for the set of all functions from A to {0,1}
Do you mean 2^A is a function, but why to {0,1}??

Actually I think it might be the case for power set, since this comes with a definition for a function:
$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$
• May 1st 2012, 07:31 AM
Plato
Re: How to understand set as exponential?
Quote:

Originally Posted by colruyt
Thanks a lot.
I don't understhand the first meaning:
Do you mean 2^A is a function, but why to {0,1}??
Actually I think it might be the case for power set, since this comes with a definition for a function:
$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$

This is confusion caused by your not posting the complete problem.
It seems that you are working of something else of which this is a sub-part.
• May 1st 2012, 07:36 AM
HallsofIvy
Re: How to understand set as exponential?
Quote:

Originally Posted by colruyt
Thanks a lot.
I don't understhand the first meaning:
Do you mean 2^A is a function, but why to {0,1}??

No, he means that 2^A is a set of functions- the set of all functions from A to {0, 1}. And it is to {0, 1} because of the "2". 3^A would be a set of all functions from A to {0, 1, 2}, or more generally, from A to any set containing 3 objects.

Quote:

Actually I think it might be the case for power set, since this comes with a definition for a function:
$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$
• May 1st 2012, 07:37 AM
colruyt
Re: How to understand set as exponential?
Quote:

Originally Posted by Plato
This is confusion caused by your not posting the complete problem.
It seems that you are working of something else of which this is a sub-part.

Sorry my mistake: basically it's a function that matches two sets into two sets, think of match students into schools. where $\displaystyle \Theta$ is the preference of students, and a typo in my previous function, the correct form is:
$\displaystyle \mu: C \cup \Theta \rightarrow 2^\Theta \cup (C \cup \Theta)$

My original guess was power set too, but I am just no sure about the notation...