Solution to (a) If A={1,2,3,4}, calculate the number of reflexive relations on A. (b) If B={1,2,3,4,5}, calculate … - Sikademy
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Archangel Macsika

(a) If A={1,2,3,4}, calculate the number of reflexive relations on A. (b) If B={1,2,3,4,5}, calculate the number of symmetric relations on B.

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(a) Reflexive relations are those where (a,a) belongs to the relation set. Now number of such elements is 4 namely, (1,1),(2,2),(3,3),(4,4). But number of elements of A\times A is 4^2=16. Hence number of reflexive relations = number of relations containing those four elements. So fixing those 4 elements we have to take subsets of 16-4=12 elements. Hence number of reflexive relations are 2^{12}.

(b) For symmetric, we need if (a,b) is there then (b,a) is also there. If a=b, then (a,b)=(b,a). Total number of elements of the form (a,b) is 5\times 5=25. So number of elements (a,b) with a\neq b is 25-5=20. We take only one element out of (a,b) and (b,a). Then there are 20/2=10 such elements. So total we get 10+5=15 elements. So number of subsets we get is 2^{15}. Now in each of these sets, we add the elements (b,a) corresponding to (a,b) and thus get all symmetric relations. Hence number of symmetric relations is 2^{15}.


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