Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
0 votes

Show that the relation \(R\) in \(R\) defined as \(R = {(a, b) : a \leq b} \), is reflexive and 
transitive but not symmetric.

Can you answer this question?

1 Answer

0 votes
  • A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$
  • A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;\Rightarrow \; (a_2,a_1)\in R \;$ for $\;a_1,a_2 \in A$
  • A relation R in a set A is called transitive. if $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R \Rightarrow \;(a_1,a_3)\in R\; $for all $\; a_1,a_2,a_3 \in A$
Given in a set of real numbers, the relation $R=\{(a,b):a\leq b\}$:
For any $a \in R$, $a \leq a$. Therefore $(a,a) \in R$. Hence $R$ is reflexive.
For any $a \neq b$, we observe that while $a \leq b$ might be true, $b \leq a$ will not be, unless $a=b$.Hence is $R$ is not symmetric.
We can verify this with a simple subsitution:
Let $a = 2, b = 4$. While $a \leq b \rightarrow 2 \leq 4$ is true, $b \leq a \rightarrow 4 \not \leq 2$. Hence $R$ is not symmetric.
For any $(a,b) \in R$, if $a \leq b$ and for any $(b,c) \in R, b \leq c$:
$\Rightarrow a \leq b \leq c \rightarrow a \leq c$.
$\Rightarrow$ If $a \leq c$ then $R$ is transitive.
answered Feb 20, 2013 by meena.p
edited Mar 8, 2013 by balaji.thirumalai

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App