# Examples of set functions

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.

• Your example is corrcet. – Mauro ALLEGRANZA Apr 19 '17 at 11:43
• Another example can be the Lebesgue measure of an interval. – Mauro ALLEGRANZA Apr 19 '17 at 11:44
• See also Probability measure. – Mauro ALLEGRANZA Apr 19 '17 at 11:45
• I have seen these functions on Wikipedia under the article set function. But as I said I've no knowledge to perceive these things at this level. So it would be highly appreciable if you give more general ones. Thanks – Abhinav Dhawan Apr 19 '17 at 11:57
• "the function that gives a set its cardinality: I could not understand how this would be a function." Consider for simplicity only the collection $\text {Fin}$ of finite sets; then the function $\text {Card} : \text {Fin} \to \mathbb N$ assign to every set $A$ exactly a (natural) number: the number $\text {Card}(A)$ of its elements. Thus, if $A=\emptyset$ (the empty set), then $\text {Card}(A)=0$ and if $A= \{ a, b , c \}$, then $\text {Card}(A)=3$. – Mauro ALLEGRANZA Apr 19 '17 at 12:14

## 1 Answer

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.
• Thanks a lot Mauro ALLEGRANZA and @gandalf61. Your answer helped me. So it's only that for a function to be a set function it only should have its domain as sets and output could be any mathematical objects ( generally numbers). OK. I would be really great dil if you could explain area being a set function more clearly as it should have infinite sets which may overlap in its domain. Am I correct. Anyways thanks a lot. – Abhinav Dhawan Apr 19 '17 at 12:56
• @AbhinavDhawan A function does not necessarily have area. – Cave Johnson Apr 19 '17 at 14:06