$\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} $

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

- 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.

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