Number of Surjections from (1,2,3,...,n) onto (1,2)

To find the number of surjections from the set (1,2,3,...,n) onto the set (1,2), we can use the principle of inclusion-exclusion.

Principle of Inclusion-Exclusion:

The principle of inclusion-exclusion is a counting technique used to calculate the number of elements in the union of multiple sets, taking into account the overlap between the sets.

Step 1: Total Number of Functions

The total number of functions that can be defined from the set (1,2,3,...,n) onto the set (1,2) is given by:

2^n

This is because for each element in the set (1,2,3,...,n), there are 2 choices to map it to in the set (1,2) - either 1 or 2. Since there are n elements in the set (1,2,3,...,n), the total number of functions is 2^n.

Step 2: Excluding Functions with Empty Images

To find the number of surjections, we need to exclude the functions that have empty images. In other words, we need to exclude the functions that do not map any element to the set (1,2).

The number of functions with empty images is 2^(n-2). This is because for each element in the set (1,2,3,...,n), there are 2 choices to map it to in the set (1,2), except for the last two elements. For the last two elements, there is only one choice - they must be mapped to the set (1,2) to ensure a surjection.

Therefore, the number of functions with empty images is 2^(n-2).

Step 3: Including Functions with One Empty Image

Next, we need to consider the functions that have exactly one empty image. This means that one element from the set (1,2,3,...,n) is not mapped to the set (1,2).

The number of functions with one empty image can be calculated using the principle of inclusion-exclusion. The number of functions with one empty image is given by:

C(n,1) * 2^(n-3)

Here, C(n,1) represents the number of ways to choose one element from the set (1,2,3,...,n), and 2^(n-3) represents the number of ways to assign the remaining elements to the set (1,2), excluding the last two elements.

Step 4: Calculating the Number of Surjections

Finally, we can find the number of surjections by subtracting the number of functions with empty images and the number of functions with one empty image from the total number of functions.

Number of surjections = 2^n - 2^(n-2) - C(n,1) * 2^(n-3)

Simplifying this expression gives us the final answer for the number of surjections from the set (1,2,3,...,n) onto the set (1,2).