When A = {1,b,ø}, how many reflexive relations exist on the set?
I have said that AxA={(1,1), (b,b), (ø,ø), (1,b), (1,ø), (b,1), (b,ø), (ø,1), (ø,b)}
Would I be right in saying that there are only 3 reflexive relations in this set? I believe that the first three elements in the set are the reflexive relations, but I am not too sure.
Furthermore, how many are both reflexive and symmetric? Is it 3 again?
A relation on $A$ is a subset of $A\times A$. A reflexive relation, in this case, is any relation $R$ such that $(1, 1) \in R$ and $(b, b)\in R$ and $(\varnothing, \varnothing) \in R$. Any subset of $A\times A$ containing those three ordered pairs is reflexive.
One such relation is given by $R_1 = \{(1, 1), (b, b), (\varnothing, \varnothing)\}$.
Another reflexive relation is $A\times A$.
There are $2^{9-3}= 2^6$ reflexive relations in all, which also gives the number of subsets of $A\times A$ containing $(1, 1), (b, b),$ and $(\varnothing, \varnothing)$.