Relations that are: reflexive but not transitive; transitive but not symmetric; symmetric but not reflexive

Question

I have an incomplete answer to my question. Can anyone help me answer the last two parts.

My question is:

Find example of a set $S$ and three relations $R_1$, $R_2$, $R_3$ on it such that

• $R_1$ is reflexive but not transitive
• $R_2$ is transitive but not symmetric
• $R_3$ is symmetric but not reflexive

My solution is:

Let's start with the first part of the question. For simplicity, we will use a small set to work with, say $\{a, b, c\}$.

First, the definitions. A binary relation (we'll call it $R$) is reflexive if $(x, x) \in R$. And a relation is transitive if $(x , y)\in R$ and $(y, z) \in R$ implies that $(x, z) \in R$.

So an example of a reflexive relation that is not transitive on $\{a,b,c\}$ would be the following relation:

$$\{(a,a),(b,b),(c,c), (a,b), (b,c)\}$$

Note that every element is in relation to itself, so it is reflexive. However, it is not transitive, because although $a$ is in relation with $ b$ and $b$ is in relation to $c$, $a$ is not in relation $c$.

Can anyone help me answer the other two parts of the question ?

Answer 1

Nicely done on the first answer. Here's a pictorial hint for the latter two: