Answer is (b)
SYMMETRIC:- Symmetric can contains both pair of elements as well as diagonal elements. e.g. (1,1) (1,2) (2,1)
ANTI-SYMMETRIC :- It contains only one set of elements from both pairs and diagonals elements. e.g. (1,1) (1,2) (2,3)
TRANSITIVE :- If the set contains elements like (1,2) and (2,3) then it should contain (1,3) as well. If not then it is fine. It accepts diagonal elements as well as other elements. e.g. (1,2) (2,3) (1,3) (1,1) (2,4)
a) R1 = { (a,a) (c,c) } is Symmetric, Anti-Symmetric and Transitive on A
This is True because this satisfies the property of all the above relations.
b) R2 = { (a,b) (b,a) (a,c) (c,a) (c,d) } is Symmetric and Anti-Symmetric
This is false coz (a,b) (b,a) are present which violates the property of Anti-Symmetric and (c,a) (c,d) are present for which (a,c) (d,c) are not present.
c) R3 = { (b,c) (c,b) (d,d) } is Symmetric but not Anti-Symmetric
This is true.
d) R4 = { (a,b) (b,c) (c,c) } is Anti-Symmetric but not Symmetric
This is true coz the order pairs sets of (a,b) is (b,a) and (b,c) is (c,b) which is not present, hence not satisfying the property of Symmetric.