IIT JEE , AIEEE Forum - Mathematics , Algebra - no of into-onto functions

magiclko

newayz m posting here too

Let the function be represented by A--> B whr n(A) = n(B)= n

total no of distinct functions from A to B = nⁿ

now no of ways in which y_i (

B) is not the image of x_i(

A) =no of into functions

Let A_i denotes the event that ith element of B is not the image of any element of A (i.e. A₁ is the event in which y₁ is nt the image of any x

A )

thrfore no of ways in which atleast one element of B is not the image of any elemnt in A , (E)= n (A₁

A₂

A₃

A₄

.......A_n)

now total no of functions when one element of B doesnt have any pre-image

= no of ways of selecting that elemnt X no of ways in which rest can have thr pre-images

=ⁿC₁ (n-1)ⁿ

similarly when 2 elemnts doesnt have pre-image then no of functions = ⁿC₂ (n-2)ⁿ

.....

thus n(E) = ⁿC₁ (n-1)ⁿ- ⁿC₂ (n-2)ⁿ + ⁿC₃ (n-3)ⁿ- ......upto n terms

this will be the no of into fucntions frm A to B

thrfore onto functions = nⁿ - n(E)