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