Question

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

The correct option is C $R_2$ is symmetric but it is not transitive.

As $(b,c)\in R_1$ but $(c,b)\notin R_1$
$\therefore R_1$ is not symmetric.
As $(b,c),(c,a)\in R_1$ but $(b,a)\notin R_1$
$\therefore R_1$ is not transitive.

We can see that $R_2$ is symmetric.
Also, $(c,a),(a,b)\in R_2$ but $(c,b)\notin R_2$
$\therefore R_2$ is not transitive .