Cardinality of a Set

1. Identify the Size of the Sets
First, determine the number of elements in both given sets:
Set $$A = \{1,2,3,4,5,6\}$$, so the number of elements in the domain is $$n = 6$$.
Set $$B = \{9,10,11,12\}$$, so the number of elements in the codomain is $$m = 4$$.

2. Apply the Onto Functions Formula
An onto (surjective) function requires that every element in the codomain has at least one corresponding element in the domain. The standard formula to find the total number of onto functions from a set of size n to a set of size m is given by the principle of inclusion-exclusion:
$$\text{Number of onto functions} = \sum_{r=0}^{m} (-1)^r \binom{m}{r} (m-r)^n$$

3. Expand and Calculate the Summation
Substitute $$n = 6$$ and $$m = 4$$ into the inclusion-exclusion formula:
$$\text{Number of onto functions} = \binom{4}{0}(4)^6 - \binom{4}{1}(3)^6 + \binom{4}{2}(2)^6 - \binom{4}{3}(1)^6 + \binom{4}{4}(0)^6$$

Evaluate each individual term step-by-step:
First term: $$\binom{4}{0}(4)^6 = 1 \times 4096 = 4096$$
Second term: $$\binom{4}{1}(3)^6 = 4 \times 729 = 2916$$
Third term: $$\binom{4}{2}(2)^6 = 6 \times 64 = 384$$
Fourth term: $$\binom{4}{3}(1)^6 = 4 \times 1 = 4$$
Fifth term: $$\binom{4}{4}(0)^6 = 1 \times 0 = 0$$

4. Compute the Final Value
Combine all the calculated terms together:
$$\text{Number of onto functions} = 4096 - 2916 + 384 - 4 + 0$$
$$\text{Number of onto functions} = 1180 + 384 - 4$$
$$\text{Number of onto functions} = 1564 - 4 = 1560$$

Consider the mapping of elements from the domain set $$A = \{1, 2, 3, 4, 5, 6\}$$ to the codomain set $$B = \{1, 2, 3, 4\}$$. We are required to determine the cardinality of the set of all surjective functions $$f: A \to B$$ subject to the explicit restriction that $$f(1) \neq f(2)$$. Let $$S$$ denote the total set of all surjective functions from $$A$$ to $$B$$, and let $$E$$ denote the subset of these surjections where the condition is violated, namely $$f(1) = f(2)$$. The number of valid functions is given by the difference $$\vert S \vert - \vert E \vert$$.

To evaluate the total number of unconstrained surjective functions $$\vert S \vert$$, we apply the Principle of Inclusion-Exclusion. The total number of unrestricted mappings from a 6-element set to a 4-element set is $$4^6$$. We must subtract the functions that omit at least one element of the codomain, add back those that omit at least two elements, and subtract those that omit three elements. This yields the classical summation:

$$\vert S \vert = 4^6 - \binom{4}{1}3^6 + \binom{4}{2}2^6 - \binom{4}{3}1^6$$

Evaluating each term explicitly gives:

$$\vert S \vert = 4096 - 4(729) + 6(64) - 4(1) = 4096 - 2916 + 384 - 4 = 1560$$

We next determine the number of invalid surjections $$\vert E \vert$$ where $$f(1) = f(2)$$. Because the elements 1 and 2 map to the exact same image in the codomain, they behave as a single entity within the domain. This equivalence collapses the effective size of the domain from 6 elements to 5 elements, transforming the problem into finding the number of onto functions from a 5-element set to a 4-element set. Applying the Principle of Inclusion-Exclusion to this modified configuration yields:

$$\vert E \vert = 4^5 - \binom{4}{1}3^5 + \binom{4}{2}2^5 - \binom{4}{3}1^5$$

Evaluating these terms explicitly gives:

$$\vert E \vert = 1024 - 4(243) + 6(32) - 4(1) = 1024 - 972 + 192 - 4 = 240$$

Finally, by subtracting the number of invalid configurations from the total unconstrained surjections, we isolate the exact number of functions that satisfy the original hypothesis:

$$\text{Valid Functions} = \vert S \vert - \vert E \vert = 1560 - 240 = 1320$$