Question:
Let R be the relation on the set A = {1, 2, 3, 4} given by
R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then,
(a) R is reflexive and symmetric but not transitive
(b) R is reflexive and transitive but not symmetric
(c) R is symmetric and transitive but not reflexive
(d) R is an equivalence relation
Solution:
(b) R is reflexive and transitive but not symmetric.
Reflexivity : Clearly, $(a, a) \in R \forall a \in A$
So, $R$ is reflexive on $A$.
Symmetry : Since $(1,2) \in R$, but $(2,1) \notin R$,
$R$ is not symmetric on $A$.
Transitivity : $S$ ince, $(1,3),(3,2) \in R$ and $(1,2) \in R$,
$R$ is transitive on $A$.