How many one-to-one correspondences exist between two sets with
A) 8 elements each?
B) n-1 elements each?
Do you mean how many separate pairings exist between two groups of objects, each of which contains eight elements? If so, the answer is (8 x 8) = 64.
The reason I'm slightly cautious about the answer is that I'd have expected the second part of the question to be asking the same about n elements each as opposed to (n-1), unless this is leading up to a simple demonstration of mathematical induction, i.e. demonstrate the answer for n=1, then prove that if it's true for (n-1) then it must be true for n.
To find the number of one-to-one correspondences between two sets, we need to use the concept of permutations. A permutation is an arrangement of objects in a specific order. In this case, the objects are the elements of the two sets.
A) If both sets have 8 elements each, we want to find the number of one-to-one correspondences between them. Since the sets have the same number of elements, we can consider one set as the "source" set and the other set as the "target" set.
To calculate the number of one-to-one correspondences, we need to find the number of permutations of the target set. Since each element in the target set can be paired with exactly one element from the source set (due to the one-to-one correspondence condition), the number of permutations of the target set will give us the total number of one-to-one correspondences.
The number of permutations of a set with 8 elements is given by 8!, which means factorial of 8. Factorial of a number n (denoted as n!) is the product of all positive integers from 1 to n.
Therefore, the number of one-to-one correspondences between two sets with 8 elements each is 8!.
B) Now let's consider the case where both sets have n-1 elements each. Similarly, we can consider one set as the source set and the other as the target set.
The same logic applies here as in part A. The number of permutations of the target set will give us the number of one-to-one correspondences.
The number of permutations of a set with n-1 elements is (n-1)!. Therefore, the number of one-to-one correspondences between two sets with n-1 elements each is (n-1)!.
Note: It is worth mentioning that the number of one-to-one correspondences will be the same regardless of the elements in the sets. It solely depends on the number of elements present in each set.