Calculating the Number of Functions
To calculate the number of functions from the set {a, b, c, d} to the set {1, 2, 3, 4}, we need to consider each element in the domain set mapping to an element in the codomain set.

Mapping for each element
- For element 'a', there are 4 choices in the codomain set. So, 'a' can map to any of the 4 elements in the codomain set.
- Similarly, 'b', 'c', and 'd' also have 4 choices each to map to elements in the codomain set.

Total number of functions
Since each element in the domain set can independently map to any element in the codomain set, the total number of functions is given by the product of the number of choices for each element.
4 choices for 'a' * 4 choices for 'b' * 4 choices for 'c' * 4 choices for 'd' = 4^4 = 256

Removing functions that are not onto
However, not all of these 256 functions are onto because some functions may not cover all elements in the codomain set. We need to remove these functions.

Calculating the number of onto functions
To calculate the number of onto functions, we can use the principle of inclusion-exclusion. The number of onto functions is given by the formula:
Number of onto functions = 4! - (4*3!) + (6*2!) - (4*1!) + 1 = 81
Therefore, the total number of functions from the set {a, b, c, d} to the set {1, 2, 3, 4} is 81.