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Let A = {1,2,3,4}. since each element of P(AxA) is subset of AxA, it is binary relation on A
Assuming each relation in P(AxA) is equally likely to be chosen,

i. what is the probability that a randomly chosen relation is reflexive
a.  1/26
b. 1/24
c. 1/26
d. 1/212
Given Ans: 1/24
ii what is the probability that a randomly chosen relation is Symmetric
a.  1/216
b. 1/24
c. 1/26
d. 1/212
Given Ans: 1/26

asked in Set Theory & Algebra by Veteran (14.7k points)   | 126 views

1 Answer

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Best answer

 Total no. of relations on a set A of cardinality n  is $2^{n^{2}}$

i) No. of reflexive relations = $2^{n^{2} - n}$

Probability of reflexive relations = $\large \frac{2^{n^{2} - n}}{2^{n^{2}}}$  = $\large \frac{2^{4^{2} - 4}}{2^{4^{2}}}$

                                             = $\large \frac{1}{16}$

                                            = $\LARGE \frac{1}{2^{4}}$

ii) No. of symmetric relations =$\large 2^{\frac{n^{2}+n}{2}}$

Probability of symmetric relations = $\LARGE \frac{2^{\frac{n^{2}+n}{2}}}{2^{n^{2}}}$ = $\LARGE \frac{2^{\frac{4^{2}+4}{2}}}{2^{4^{2}}}$

                                                 =$\LARGE \frac{1}{64}$

                                                = $\LARGE \frac{1}{2^{6}}$

answered by Active (1.8k points)  
selected by
Perfect!!
great answer Rahul :) can u please tell us the formula for remaining properties also ( transitive, antisymm) also thanks :)

Anti Symmetric = $\LARGE 2^{n}*3^{\frac{n^{2}-n}{2}}$

Asymmetric = $\LARGE 3^{\frac{n^{2}-n}{2}}$

No formula exists for Transitive Relations!



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