#relation closed

Question

The Number of Relations, Which are both Reflexive and Symmetric but not Anti-Symmetric, on a

set with 6 elements, are ____________?

Comments

32768?

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2^15-1??

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@Soumya Tiwari

it should be 32767 ,you also have to subtract the case when only diagonal elements are present!

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Thanks got your point :)

Answer 1

There is only 1 relation which is reflexive, symmetric and anti-symmetric

Number of symmetric relations which are reflexive = $\large 2^{\frac{n(n-1)}{2}} = $$\large 2^{\frac{6(5)}{2}} = 32768$

Total number of symmetric relations which are reflexive but not anti-symmetric = $32768 - 1 = 32767$

 

Comments

There is only 1 relation which is reflexive, symmetric and anti-symmetric

How?

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See the examples below

(using elements)={relations possible which is reflexive , symmetric and anti symmetric}

{0,1} ={(0,0),(1,1)}

{0,1,2}={(0,0),(1,1),(2,2)}

{0,1,2,3}={(0,0),(1,1),(2,2),(3,3)}

{0,1,2,3,...,n}={(0,0),(1,1),(2,2),(3,3),...,(n,n)}

and so on.

It  is clear that the given relations are reflexive and if we remove any one ordered pair from RHS then the relation would no longer be reflexive. Also ,if we add more ordered pair then the relations will remain reflexive.

 for symmetric relation if (a,b) is present then (b,a) should also be present

 for anti symmetric relation if (a,b) is present then (b,a) should not be present but if there is an exception that ordered pair like (a,a),(b,b)....(n,n) could be present.

So in the above example if we add any ordered pair like (a,b) (lets say, (1,3) in third example) then we have to add (b,a)(i.e. (3,1)) also to make it symmetric and this would violate the anti symmetric property.

on the other hand if we add any ordered pair like (a,b) (lets say, (1,3) in third example) then we do not  add (b,a)(i.e. (3,1)) the relation would no longer be symmetric but would still satisfy the anti symmetric property.

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@Balaji Jegan here's how

https://math.stackexchange.com/questions/2164978/how-many-reflexive-symmetric-and-antisymmetric-relations-are-there-on-an-n-ele

See bof comment.