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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 is 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
(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 So number of elements (a,b) with 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 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