Let $S=\{a_{1},a_{2},...,a_{n}\}$
(i)Find number of binary relations $(*)$ on $S$ which are commutative.
(ii)Find number of binary relations on $S$ such that $a_{i}*a_{j}\neq a_{i}*a_{k},\forall j\neq k.$
(iii)Let $a_{1},a_{2},...,a_{n}$ be the distinct real numbers,then find total number of binary relations on $S$ such that $a_{i}*a_{j}\leq a_{i}*a_{j+1}\forall i,j$
My Attempt:
(i)Total number of pairs $(a_{i}*a_{j})$ possible are $\binom{n}{2}+n$ i.e.$\frac{n^2+n}{2}$. Each pair has $n$ choices. So number of binary relations should be $n^{\left(\frac{n^2+n}{2}\right)}$.
But how many of these will be commutative.
(ii)Now, $(a_{i}*a_{1})$ has $n$ choices so $(a_{i}*a_{2})$ has $(n-1)$ choices and ....so on. So $(a_{i}*a_{j})$ has $n!$ choices. So for all $i$ required number of binary relations should be $(n!)^n$
(iii)How to define $*$ if $\leq$ is to be invoked.