Number of Ordered Pairs (A, B) where A and B are subsets

To find the number of ordered pairs (A, B) where A and B are subsets of {1,2,3,4,5} such that neither A is a subset of B nor B is a subset of A, we can use the principle of inclusion-exclusion.

1. Total number of subsets:
The set {1,2,3,4,5} has 2^5 = 32 subsets. Each element in the set can either be present or absent in a subset.

2. Number of pairs where A is a subset of B:
For A to be a subset of B, all the elements of A must also be present in B. Therefore, for each element in the set, we have 2 choices - either include it in B or exclude it from B. So, the number of pairs where A is a subset of B is 2^5 = 32.

3. Number of pairs where B is a subset of A:
Similar to the previous case, for each element in the set, we have 2 choices - either include it in A or exclude it from A. So, the number of pairs where B is a subset of A is also 2^5 = 32.

4. Number of pairs where both A and B are subsets of each other:
If A is a subset of B and B is a subset of A, then A and B must have the same elements. In other words, A and B must be equal sets. There is only one such set, which is the empty set {}. Therefore, there is only 1 pair where both A and B are subsets of each other.

5. Number of pairs where neither A is a subset of B nor B is a subset of A:
To find this, we subtract the number of pairs where A is a subset of B, the number of pairs where B is a subset of A, and the number of pairs where both A and B are subsets of each other from the total number of subsets.

Total number of pairs = 2^5 - 32 - 32 + 1 = 32 - 32 + 1 = 1

Therefore, there is only 1 ordered pair (A, B) where A and B are subsets of {1,2,3,4,5} such that neither A is a subset of B nor B is a subset of A.