Abstract Algebra topic: Equivalence relations

Question

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.

Answer 1

$R_1$ 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.

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For the specific $S$ the relations could be $$\begin{align*}R_1 = & \{(1,1), (1,2), (1,3), (2,1), (3,1) , (2,2), (3,3)\} & = \{1\} \times S \cup S \times \{1\} \cup {\rm id}\\ R_2 = & \{(1,2), (2,3), (1,3)\} \\ R_3 = & \{(1,2), (1,3), (2,3), (2,1), (3,1), (3,2)\} \end{align*}$$

Answer 2

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. $$x \sim y :\Leftrightarrow x \equiv y \quad ({\rm Mod}\ n)$$