# 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

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.