Recall that a relation on a n-element set can be represented by a n x n matrix of 1 s and 0 s. Such a matrix has n^(2) entries in total, with n diagonal entries and n^(2)-n off-diagonal entries.
For each of the counting questions below, give a combinatorial expression in terms of n along with a 1-2 sentence explanation. On a n-element set, how many different relation matrices are...
1. irreflexive?
2. symmetric?
3. connected?
4. irreflexive or symmetric?
5. irreflexive or connected?
6. symmetric or connected?