Thread: How to understand set as exponential?

1. 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.

2. Re: How to understand set as exponential?

It's the set of all functions from theta to {0,1}.

3. Re: How to understand set as exponential?

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 $\displaystyle A$.

4. Re: How to understand set as exponential?

Thanks a lot.
I don't understhand the first meaning:
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:
$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$

5. Re: How to understand set as exponential?

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.

6. Re: How to understand set as exponential?

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.

Actually I think it might be the case for power set, since this comes with a definition for a function:
$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$

7. Re: How to understand set as exponential?

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...