# Given a relation $$R$$ in $$R$$ defined as $$R = {(a, b) : a \leq b}$$, is this reflexive,  transitive or symmetric.

$\begin{array}{1 1} \text{Reflexive, Transitive and Symmetric} \\ \text{Reflexive and Transitive but not Symmetric} \\ \text{Not Reflexive but Transitive and Symmetric} \\ \text{Neither Reflexive, Transitive or Symmetric} \end{array}$

Toolbox:
• 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.