Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Question:
Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Solution:
$R=\{(a, b) ; a \leq b\}$
Clearly $(a, a) \in R$ as $a=a$.
∴R is reflexive.
Now,
$(2,4) \in R(\operatorname{as} 2<4)$
But, $(4,2) \notin R$ as 4 is greater than 2 .
$\therefore R$ is not symmetric.
Now, let $(a, b),(b, c) \in R$.
Then,
$a \leq b$ and $b \leq c$
$\Rightarrow a \leq c$
$\Rightarrow(a, c) \in R$
$\therefore R$ is transitive.
Hence,R is reflexive and transitive but not symmetric.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.