Suppose that A has exactly three elements and B has exactly two. How many different functions are there from A to B? How many of these are injective? How many are surjective?
9,6,0
To determine the number of different functions from set A to set B, we need to consider the possible assignments of elements from A to elements in B.
Given that A has exactly three elements and B has exactly two elements, let's consider the possibilities for each element in A:
For the first element in A, there are two choices, as it can be assigned to any element in B.
For the second element in A, there are also two choices, as it can be assigned to any element in B.
For the third element in A, there are again two choices, as it can also be assigned to either element in B.
Since each of the three elements in A has two choices, we can multiply the number of choices for each element together to determine the total number of different functions:
Number of different functions from A to B = 2 * 2 * 2 = 8.
Now, let's consider injective functions. An injective function is one in which distinct elements in A get mapped to distinct elements in B.
For the first element in A, there are two choices, as it can be assigned to either element in B.
For the second element in A, there is only one choice remaining, as one element in B has already been assigned.
For the third element in A, there is only one choice remaining as well, as both elements in B have been assigned.
Again, since each of the three elements in A has a different number of choices, we can multiply the number of choices for each element together to determine the total number of injective functions:
Number of injective functions from A to B = 2 * 1 * 1 = 2.
Lastly, let's consider surjective functions. A surjective function is one in which every element in B is assigned to by at least one element in A.
Since B has exactly two elements, and A has three elements to choose from, we can see that it is not possible to have a surjective function from A to B. This is because there will always be one element in A that has no corresponding element in B.
Therefore, the number of surjective functions from A to B is 0.
To summarize:
- The number of different functions from A to B is 8.
- The number of injective functions from A to B is 2.
- The number of surjective functions from A to B is 0.