Relations

(i) Identify the relation which is reflexive, transitive but not symmetric:
- Reflexive: A relation is reflexive if for all , .
- Transitive: A relation is transitive if for all and , .
- Symmetric: A relation is symmetric if for all , .
Now, let's check each relation for reflexivity, transitivity, and symmetry.
- :
- Not reflexive (missing , , ).
- Not transitive because we don't have or for transitivity.
- Not symmetric since is in , but is not.
- :
- Not reflexive (missing , ).
- Transitive: It is transitive since if we have and , we also have , and similarly for other combinations.
- Not symmetric, because is in , but is not.
- :
- Reflexive because it includes , , and .
- Symmetric because for every pair , is also present.
- Not transitive because there is no , which makes it not transitive.
Thus, is reflexive and symmetric but not transitive.
(ii) Identify the relation which is reflexive and symmetric but not transitive:
From the analysis above, is reflexive and symmetric but not transitive.
(iii) Identify the relations which are symmetric but neither reflexive nor transitive:
- is symmetric but neither reflexive nor transitive.
- is symmetric but neither reflexive nor transitive.
OR
(iii) What pairs should be added to the relation to make it an equivalence relation? To make an equivalence relation, it needs to be reflexive, symmetric, and transitive. - Reflexive: We need to add , , and . - Symmetric: Add and (since , are already present, but their reverse pairs are missing). - Transitive: Ensure the necessary transitive pairs are present, which may be covered after adding the missing symmetric pairs. Thus, the pairs to be added are: