11 For any relation R sR and tR denote the symmetric and tra

11. For any relation R, sR and tR denote the symmetric and transitive closures of R respectively, and |R| denotes the number of ordered pairs in R. When R = {(a,b), (a,c)} the value of |s(tR)| + |t(sR)| is (a) 8. (b) 10. (c)13. (d) 18.

Solution

|R| = no of ordered pairs in R

R = {(a,b), (a,c)}

sR being symmetric closure of R should make R a symmetric set

In other words consider sR=

{(a,b) (b,a) (a,c) (c,a)} This type of additions to R make R symmetric and hence called sR

Note that |sR| =4

If sR is to be made transitive also, then as (a,b) (b,a) are there in sR we must add (a,a) and the same applies for (a,c) (c,a)

Thus t(sR) = {(a,b) (b,a) (a,c) (c,a(a,a))}

| t(sR) |= 5

-------------------------------------------------------------------------

Consider tR, this is the relation arrived from S to make S transitivie

HencetR = {(a,b), (a,c), (b,c)(c,a) (a,a)}

To make this symmetric

s(tR) = {{(a,b), (a,c), (b,c)(c,a) (a,a), (b,a), (c,b)(b,b)}

--------------------------------------------------------------

|s(tR)| =8

NOw we find that sum = 5+8 =13