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