Identify whether the following are true or false. Explain why!
(i) On a set A = {a, b, c}, the relation R = {(a, a), (b, b), (c, c), (a, b), (b, a), (a, c)} is an equivalence relation. Check each property.
(ii) On a set A = {a, b, c}, the relation R = {(a, a), (b, b), (c, c), (a, b), (b, c), (a, c)} is a partial ordering. Check each property.
(b) Let S = {1, 2, 3, 4}.
- Given an example of a relation that is symmetric and transitive, but not antisymmetric.
- Given an example of a relation that is reflexive and antisymmetric, but not transitive.