Suppose that A 1 2 3 B 4 5 C 6 7 8 R 1 7 3 6 3 7 and S

Suppose that A = {1, 2, 3}, B = {4, 5}, C = {6, 7, 8}, R = {(1, 7), (3, 6), (3, 7)}, and S = {(4, 7), (4, 8), (5, 6)}. Note that R is a relation from A toC and S is a relation from B toC. Find the following relations:

(b) R1 S

Solution

soln.if R ₌ { (a, b) ; a,b € A } is arelation on set A. Then R^-1₌ { (b,a) ; (a,b) € R } is an inverse relation on set A.

Here R ={ (1,7), (3,6), (3,7) } is relation from A to C. Then R^-1₌{ (7,1) ,(6,3) ,(7,3) } is inverse relation from C to A.

S={ (4,7), (4,8) ,(5,6) }

To find R^-1_*S .It is a composition of 2 relations

(R-1 * S)(x) = R^-1(S(x))

Here (R^-1_*S)(4) =R-1 (S(4))= R^-1(7) =1,3 ( Becoz S(4)=7,8, but here 8 is not in the domain of R^-1 & R^-1(7) =1 &3

(R^-1_*S)(5) = R-1 (S(5)) = R^-1(6) = 3 (Becoz S(5)=6 & R-1 (6) =3

thus R^-1_*S ={ (4,1) ,(4,3) ,(5,3)}