Question

If X is a set containing n elements:

1. How many functions are there f : X -> X?

2. How many relations are there R : X -> X?

Answer 1

Ques 1)

Number of functions from X to X are:
`n^n`

Ques 2)

Number of relation from X to X are:
`2^(n^2)`

Explanation:

Relation--

A relation from a set A to set B is the set of all the ordered of the type:

(a,b) where a belongs to A and b belongs to B.

if A has m elements and B has n elements.

Then the total number of relation from A to B are:
`2^mn`

Function--

A function from a set A to set B is the set of all the ordered pair where each element of set A is mapped to a single element of the other set.

i.e. every element has a single image.

if A has m elements and B has n elements.

Then the total number of functions from A to B are:
`n^m`