GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
185 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 (30k points)  
recategorized ago by | 185 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 Jun 2017
  1. Bikram

    3704 Points

  2. Arnab Bhadra

    1502 Points

  3. Hemant Parihar

    1502 Points

  4. Niraj Singh 2

    1481 Points

  5. junaid ahmad

    1432 Points

  6. Debashish Deka

    1384 Points

  7. Rupendra Choudhary

    1220 Points

  8. rahul sharma 5

    1220 Points

  9. Arjun

    1168 Points

  10. srestha

    1010 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 Jun 26 - Jul 02
  1. Arjun

    208 Points

  2. akankshadewangan24

    152 Points

  3. Debashish Deka

    138 Points

  4. Hira Thakur

    130 Points

  5. Soumya29

    106 Points


23,399 questions
30,111 answers
67,490 comments
28,426 users