Question

Let $S$ be a finite set containing $n$ elements. Then the total number of commutative binary operation on $S$ is

• A. $n^{\left[\frac{n\left(n+1\right)}{2}\right]}$
• B. $n^{\left[\frac{n\left(n-1\right)}{2}\right]}$
• C. $n^{\left(n^2\right)}$
• D. $2^{\left(n^2\right)}$

Answer 1

Answer: (A)

Let S = {ai } where i = 1.2.....n
From commutative operations,
ai *a j = a j *ai … (i) $\forall$i, j = 1,2,3....n
where * represents a binary operation
$\therefore$ Number of distinct elements in S × S
i.e.,$\begin{array}{c}\left\{a_i\right\}\times \left\{a_j\right\}\\ i=1,\mathrm{2...}nj=1,\mathrm{2...}n\end{array}$ subject to the condition (i)
= n{(a${}_1$,a${}_1$),(a${}_1$,a${}_2$ )......(a${}_1$,an ),
(a${}_2$, a${}_2$ ), (a${}_2$,a${}_3$),....(a${}_2$,a${}_n$ ),
...(a${}_{n-1}$,a${}_{n-1}$),(a${}_{n-1}$,a${}_n$ ),(a${}_n$ ,a${}_n$ )
$=n+\left(n-1\right)+\left(n-2\right)+....+2+1=\frac{n\left(n+1\right)}{2}$
$\therefore$ No. of commutative binary operations
= No. of functions f : S × S $\to$ S subject to (i)
$=n.n.n....\frac{n\left(n+1\right)}{2}times=n^{\frac{n\left(n+1\right)}{2}}$