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Consider the set S = {a, b} and ‘L’ be a binary relation such that L = {all binary relations except reflexive relation set S}. The number of relation which are symmetric _______.

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Best answer
8 votes
8 votes
Number of symmetric relations which are not reflexive

{ }

{(a,a)}

{(b,b)}

{(a,b),(b,a)}

{(a,a),(a,b),(b,a)}

{(a,b),(b,a),(b,b)}

Hence I think 6 should be the correct ans.
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5 votes
5 votes
Since it is small set,it can be done in hit and trial.But for more elements we should do it systematically.

We need to find number of relations which are symmetric but not reflexive

= [Total Symmetric] - [Symmetric and reflexive]

=[ $2 ^ {n(n+1)/2 }$] - [1*$2 ^ {n(n-1)/2 }$  ]

= 8-2

=6
1 votes
1 votes

L  =  {all  binary relations  except  reflexive  relation  set  S}

=> L has no xRx pair

=> So no diagonal element present.

=> Total size of L = 2^(n2 - n)

In symmetric relations, xRy and yRx should both come. For each x and y (x!=y), either (x,y) or (y,x) can be present.

Number of symmetric relations in L = (No diagonal element selected)*(One of the non-diagonal pairs)

No of non-diagonal pairs =  (n2 - n)/2 

=> Number of symmetric relations =  2^((n2 - n)/2)

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