The number of reflexive relations on an n-element set is 2n2−n
Can anybody tell me why so?
Hint Every relation $\displaystyle R$ on $\displaystyle A=\{a_1,\ldots,a_n\}$ is an element of $\displaystyle \mathcal{P}( A\times A)$ . So, we have $\displaystyle \textrm{card}\;\mathcal{P}( A\times A)=2^{\;\textrm{card }\mathcal{P}( A\times A)}}=2^{n^2}$ relations on $\displaystyle A$ . Could you continue?
total elements in that matrix=nXn
total diagonal element=n
so obviously power set off-diagonal elements=2^(n^2-1).
But i have problem with the approach u are using .Can please how off-diagonal elements are correlated with reflexive relation..
Thanx for help.
Denote $\displaystyle \Delta$ the diagonal of $\displaystyle A\times A$ with $\displaystyle A=\{a_1,\ldots,a_n\}$ . Every reflexive relation $\displaystyle R$ on $\displaystyle A$ has the form $\displaystyle R=\Delta\cup B$ with $\displaystyle B\subset A\times A-\Delta$ . But $\displaystyle A\times A-\Delta$ has $\displaystyle n^2-n$ elements, so there are $\displaystyle 2^{n^2-n}$ reflexive relations on $\displaystyle A$ .