GATE CSE
First time here? Checkout the FAQ!
x
+3 votes
292 views
The number of possible commutative binary operations that can be defined on a set of $n$ elements (for a given n) is ___________.
asked in Set Theory & Algebra by Veteran (32.8k points)  
recategorized by | 292 views

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.

1 Answer

+5 votes
Best answer

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 (68.7k points)  
selected by
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..


Top Users Aug 2017
  1. ABKUNDAN

    4658 Points

  2. Bikram

    4032 Points

  3. akash.dinkar12

    3136 Points

  4. rahul sharma 5

    2856 Points

  5. manu00x

    2664 Points

  6. makhdoom ghaya

    2380 Points

  7. just_bhavana

    2040 Points

  8. Tesla!

    1756 Points

  9. pawan kumarln

    1574 Points

  10. learner_geek

    1558 Points


24,878 questions
31,952 answers
74,105 comments
30,065 users