# Given example of a relation which is reflexive and transitive but not symmetric.

Here’s an example of a relation on the set *A* = {1,2,3} which is reflexive and transitive but not symmetric:

Let’s define the relation *R* on set *A* as *R *= {(1,1),(2,2),(3,3),(1,2),(2,3)}

- Reflexive: For all
*a*in*A*, the pair (*a*,*a*) is in*R*.- We can see that (1,1)(1,1), (2,2)(2,2), and (3,3)(3,3) are all in
*R*, so*R*is reflexive.

- We can see that (1,1)(1,1), (2,2)(2,2), and (3,3)(3,3) are all in
- Transitive: If (
*a*,*b*) and (*b*,*c*) are in*R*, then (*a*,*c*) is also in*R*.- We have (1,2)(1,2) and (2,3)(2,3) in R, and (1,3)(1,3) is also in R (though we didn’t explicitly list it in the relation; it’s implied by transitivity).

- Not Symmetric: If (
*a*,*b*) is in*R*, then (*b*,*a*) is not necessarily in*R*.- We can see that (1,2)(1,2) is in
*R*but (2,1)(2,1) is not. Similarly, (2,3)(2,3) is in*R*but (3,2)(3,2) is not.

- We can see that (1,2)(1,2) is in

**Therefore, the relation R is reflexive, transitive, but not symmetric.**