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The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given n) is ___________.
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binary operation is commutative if changing the order of the operands does not change the result.

Example :

Let  be a set and  be a binary operation on

Then, * is said to be commutative if, for every x,y,z in , the following identity holds:

x * y = y * x

Here we change the order of operands x and y  but result is same

If the above equation holds for particular values of x and y, we say that  x and  y commute.

Given , the cardinality of set = n

So consequently ,

No of entries in operation table(Cayley table)  =  n2

And hence if we consider lower triangular or upper triangular half , we have : (n2 + n) / 2

And in an operation table , each entry can be filled in n ways by any one element out of given n elements of the set..

So no of ways we can fill the upper or lower triangular half  =  n(n^2 + n)/2

So each of these is nothing but an instance of operation table of commutative operation as say (i,j) entry is filled in the table so (j,i) entry will also be the same hence the choice for (j,i) entry is constrained to 1 as we are concerned about commutativ operation table here..

Therefore,

No of possible binary operations which are commutative  = n(n^2 + n)/2

answered by Veteran (89.7k points) 15 58 296
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is it equal to symetric = A*B= B*A
Ya commutative means A * B = B * A as mentioned in the answer..
@HabibKhan: i have one doubt,,no. of symmetric relations possible in a set of n elements is$2^{(n2+n)/2}$

please correct me if i am wrong..
Ya it is fine ..But here the question is about no of binary operations which are commutative..
but isn't it right that no. of lower triangular entries =(n^2-n)/2
Ya it is true ..But then no problem..