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.

Re: How to understand set as exponential?

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

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 $\displaystyle A$.

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:

$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$

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.

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:

$\displaystyle \mu: C\cup U \rightarrow 2^\Theta \cup (C\cup \Theta)$

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...