Write the formula to find the number of onto functions from set A to set B.

Solution:

Functions are the backbone of advanced mathematics topics like calculus. Functions are of many types, like into and onto. Let's solve a problem regarding onto functions.

To find the number of onto functions from set A (with m elements) and set B (with n elements), we have to consider two cases:

⇒ One in which m ≥ n: In this case, the number of onto functions from A to B is given by:

→ Number of onto functions = n^m - ⁿC₁(n - 1)^m + ⁿC₂(n - 2)^m - ....... or as [summation from k = 0 to k = n of { (-1)^k . ⁿC_k . (n - k)^m }].

Let's solve an example.

→ Let m = 4 and n = 3; then using the above formula, we get 3⁴ - ³C₁(3 - 1)⁴ + ³C₂(3 - 2)⁴ = 81 - 48 + 3 = 36.

Hence, they have 36 onto functions.

⇒ One in which m < n: In this case, there are no onto functions from set A to set B, since all the elements will not be covered in the range function; but onto functions from set B to set A is possible in this case though.

Hence, The formula to find the number of onto functions from set A with m elements to set B with n elements is n^m - ⁿC₁(n - 1)^m + ⁿC₂(n - 2)^m - ....... or [summation from k = 0 to k = n of { (-1)^k . ⁿC_k . (n - k)^m }], when m ≥ n.

Write the formula to find the number of onto functions from set A to set B.

Summary:

The formula to find the number of onto functions from set A with m elements to set B with n elements is n^m - ⁿC₁(n - 1)^m + ⁿC₂(n - 2)^m - ... or [summation from k = 0 to k = n of { (-1)^k . ⁿC_k . (n - k)^m }], when m ≥ n.