The question is to find an example of a set $S$ and three relations $R_1$, $R_2$, and $R_3$ on it, such that $R_1$ is reflexive but not transitive, $R_2$ is transitive but not symmetric and $R_3$ is symmetric but not reflexive.
This is what I have done: Let set $S=\{a,b,c,d\}$ and the three realations are as follows:
- $R_1 = \{(a,a),(b,b),(c,c),(d,d)\}$
- $R_2 = \{(a,b),(b,c),(c,d)\}$
- $R_3 = \{(a,b),(b,a),(c,d),(d,c)\}$
If there is a set which is wrong and not satisfying what is needed can anyone help?
$R_1$ is reflexive, but it is also transitive. To see that it is transitive, let $(x, y), (y, z) \in R_1$. By your definition of $R_1$, $x = y$ and $y = z$. Therefore $(x, z) = (x, y) = (y, z) = (y, y) \in R_1$. To fix this, add a pair of elements of the form $(x, y)$ and $(y, z)$ with $x \neq y$, $y \neq z$, and $x \neq z$ but do not add $(x, z)$; this breaks transitivity.
$R_2$ is not symmetric, but it is not transitive (on any pair of elements). For example, $(a, b), (b, c) \in R_2$ but $(a, c) \not\in R_2$. Try adding in the necessary elements, for example $(a, c)$. Note, after including the first lot of elements you may need to then reexamine your relation and see if there is anything else that needs to be added so that your relation is transitive.
$R_3$ is symmetric but not transitive as desired. Interestingly, a symmetric, non-reflexive relation will also be non-transitive. Can you see why?