Relations

Question

I'm getting 384....

Comments

@Pawan,

i think it should be 392

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got my mistake  joshi_nitish  sir ...thanks

Answer 1

Answer is 392.

Total relations = 2⁹ = 512

Relations which are neither Ref, nor Symmetric means we have to do Total Realtions - (Symmetric U reflexive)

Number of Reflexive relations = We have choice only on non-diagonal elements hence 2^{n^2-n} = 2^9-3 = N(R) = 2⁶

Number of symmetric relations = We have choice for diagonal elements as well as symmetric pairs = 2ⁿ * 2^{(n^2-n)/2} = N(S) = 2³ * 2³ = 2⁶

Number of relations which are both symmetric as well as Reflexive = only choice about pairs (Diagonal elements must be there) = N(R $\cap$ S) = 2^{(n^2-n)/2}  = 2³

Hence N(R U S) = N(R) + N(S) - N(R $\cap$ S)

= 64 + 64 - 8 = 120

Hence niether Ref, nor symmetric = 512-120 = 392.