Question

Let $A=\{1,2,3\}$ and let

$R_1=\{(1,1),(1,3)(3,1)(2,2),(2,1)(3,3)\}$

$R_2=\{(2,2),(3,1),(1,3)\}$ $R_3=\{(1,3),(3,3)\}$

$R_4=A\times A.$

Find which of the given each of the relations $R_1,R_2,R_3,R_4,$ on A satisfy all the given properties:

(a) reflexive (b) symmetric (c) transitive

• A. $R_1$
• B. $R_2$
• C. $R_3$
• D. $R_4$

Answer 1

The correct option is D $R_4$

Given $A=\{1,2,3\}$
Let $R_1=\{(1,1),(1,3)(3,1)(2,2),(2,1)(3,3)\}$
$R_2=\{(2,2),(3,1),(1,3)\}$
$R_3=\{(1,3),(3,3)\}$
$R_4=A\times A$
So, $R_4=\left\{\right(1,1\left)\right(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3\left)\right\}$
Clearly, $R_4$ is reflexive , symmetric, transitive as it contains all the ordered pairs.
Now, we will check about $R_1$
$R_1$ is reflexive on $A$ as every element of set $A$ is related with itself.
$R_1$ is not symmetric as $(2,1)\in R$ but $(1,2)\notin R$
$R_1$ is not transitive as $(2,1),(1,3)\in R$ but $(2,3)\notin R$
Next ,we will check about $R_2$
$R_2$ is not reflexive because $(1,1),(3,3),(4,4)\notin R$
$R_2$ is symmetric because $(1,3),(3,1)\in R$
$R_3$ is not transitive as $(1,3),(3,1)\in R$ but $(1,1)\notin R$
Now, we will check about $R_3$
$R_3$ is not reflexive because $(1,1),(2,2),(4,4)\notin R_3$
$R_3$ is not symmetric because $(1,3)\in R$ but $(3,1)\notin R_3$
$R_3$ is transitive as $(1,3),(3,3)\in R$ so, $(1,3)\in R_3$