GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
157 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 Combinatory by Veteran (29.5k points)   | 157 views

1 Answer

+3 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 (66.5k 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 May 2017
  1. akash.dinkar12

    3154 Points

  2. pawan kumarln

    1630 Points

  3. sh!va

    1590 Points

  4. Arjun

    1350 Points

  5. Devshree Dubey

    1246 Points

  6. Angkit

    1044 Points

  7. Debashish Deka

    1022 Points

  8. Bikram

    972 Points

  9. LeenSharma

    836 Points

  10. Prashant.

    692 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. pawan kumarln

    256 Points

  2. Ahwan

    236 Points

  3. jjayantamahata

    114 Points

  4. joshi_nitish

    114 Points

  5. Arnab Bhadra

    94 Points


22,731 questions
29,061 answers
65,101 comments
27,627 users