Question

Consider the following two binary relations on the set $A = \{a, b, c\} : R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$ and $R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a),(b, b), (a, c)\}.$ Then :

• A. both $R_1$ and $R_2$ are not symmetric.
• B. $R_1$ is not symmetric but it is transitive.
• C. $R_2$ is symmetric but it is not transitive.
• D. both $R_1$ and $R_2$ are transitive.

Answer 1

Answer: (C)

$R_{1}=\{(c, a),(b, b),(a, c),(c, c),(b, c),(a, a)\}$
$b, c \in R_{1}$ c, a $\notin R_{1} R_{1}$ is not symmetric $(b, c),(c, a) \in R_{1}(b, a) \notin R_{1}, R_{1}$ is not transitive
$R _{2}=\{( a , b ),( b , a ),( c , c ),( c , a ),( c , a ),( a , a ),( b , b ),( a , c )\}$
$\forall(a, b) \in R_{2}(b, a) \times R_{2}$
Therefore it is symmetric
$(c, a),(a, b) \in R_{2}(c, b) \notin R_{2}$
Therefore $R _{2}$ is not transitive