I have recently got acquainted with a special kind of function known as a set function . I've a series of questions in my mind with respect to this .
Firstly it is hard on my part, at this level to have an understanding of set function. It is defined as a function which takes an input a set and gives a number as output. First of all I'm not able to grasp how a set can be taken as input and above that how can it give a number as a output. So I want anyone to explain me clearly what a set function is and how it works.
The next thing is that I'm unable to find examples of a set function . One example that I could partially understand is that the function that gives a set its cardinality is a set function . I said that I could partially relate to this is as I could not understand how this would be a function. I also searched this on Wikipedia but the examples they gave were beyond my level of understanding. So I would be highly thankful if someone give me examples of set function but I don't want complicated one's which are beyond my thinking.
Lastly I want to ask how area is a set function and does it have infinite sets as domain. In general I want to know how area is a set function .
Thanks in advance for any possible help.
$\endgroup$ 61 Answer
$\begingroup$A set function is simply a rule that assigns a mathematical object (the output) to each set (the input). In most of the examples of set functions the input is a set of real numbers or points in $R^n$and the output is a single real number, but they do not have to be.
Some simple examples of set functions are:
- A function that assigns the number 0 to each set. We could call this the "zero function".
- A function that assigns the number 1 to a set if it contains the word "zebra", and assigns the number 0 otherwise. We could call this the "characteristic function" for the word "zebra".
- A function that assigns each set to itself. We could call this the "identity function".
- A function that assigns the integer n to a set if the set can be out in one-to-one correspondence with the set of integers {1,2,3...n}. This is similar to the "cardinality" function, but as it stands it only works for finite sets. To make it a true set function we would have to extend it to assign an output to infinite sets as well.
