Let B={1,2,3.....m), let R be a relation on B and its antisymmetric

a) What is the biggest number of ordered pairs that can be in R?

b)How many antisymmetric relations on A have the size found in a) ??

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- Nov 26th 2009, 04:52 PMscubasteve123relations
Let B={1,2,3.....m), let R be a relation on B and its antisymmetric

a) What is the biggest number of ordered pairs that can be in R?

b)How many antisymmetric relations on A have the size found in a) ?? - Nov 26th 2009, 08:04 PMShanks
(a)Since the matrix representation M of R can be a upper triangle matrix, it has the bigggest oder $\displaystyle \frac{n(n+1)}{2}$.

(b)Since the element in the diagonal of M should be 1, Thus there are $\displaystyle 2^{\frac{n(n-1)}{2}}$ distinct antisymmetric relations on B having the size found in (a) - Nov 27th 2009, 03:19 AMemakarov
The same problem was also considered here. If you have difficulties, I suggest starting with writing some examples of such relations with the maximum number of pairs for small $\displaystyle m$.