Abstract Algebra topic: Equivalence relations If R1 is reflective and not transitive, R2 is transitive but not symmetric and R3 is symmetric but not reflexive. We need to find an example of a set S and the three relations R1 R2 R3.
$R_1$ use the $\delta_{1}$ relation "made" reflective ($=1 \vee =$)
$R_2$ use the implication relation ($\Rightarrow$) in any fashion;
$R_3$ use the inequality relation ($\neq$) in any fashion.
$S$ could be $\{1,2,3\}$ for example.



For the specific $S$ the relations could be
$$\begin{align*}R_1 = &amp; \{(1,1), (1,2), (1,3), (2,1), (3,1) , (2,2), (3,3)\} &amp; = \{1\} \times S \cup S \times \{1\} \cup {\rm id}\\
R_2 = &amp; \{(1,2), (2,3), (1,3)\} \\
R_3 = &amp; \{(1,2), (1,3), (2,3), (2,1), (3,1), (3,2)\}
\end{align*}$$
Suppose $S$ is the set of all integers and $n$ be some fixed integer. Take the relation on $S$, that $x$ is related to $y$ if $y$ when divided by $n$ leaves the same remainder as $x$ leaves when divided by $n$. i.e.
$$x \sim y :\Leftrightarrow x \equiv y \quad ({\rm Mod}\ n)$$