Let the set 𝐴={1,2,3,4}. and consider the following relations on the set A R={(1,1),(2,2),(3,3),(4,4)} S={(1,2),(2,1),(3,4),(4,3), (3,3)} T={(1,2),(2,3),(1,3), (4,4)} U={(1,3),(2,4),(3,1),(4,2),(1,1),(1,4)}then:R is and  and S is and  and T is and  and U is and  and

The relations R, S, T, and U are defined on the set A={1,2,3,4}. We can determine the properties of these relations by examining their ordered pairs.

1. Relation R={(1,1),(2,2),(3,3),(4,4)}:

   • R is reflexive because every element in set A is related to itself.
   • R is not symmetric because there are no pairs (a,b) and (b,a) for a≠b.
   • R is not transitive because there are no pairs (a,b), (b,c) and (a,c) for a≠b≠c.
   • R is not antisymmetric because there are no pairs (a,b) and (b,a) for a≠b.

2. Relation S={(1,2),(2,1),(3,4),(4,3), (3,3)}:

   • S is not reflexive because not every element in set A is related to itself.
   • S is symmetric because for every pair (a,b), there is a pair (b,a).
   • S is not transitive because there are no pairs (a,b), (b,c) and (a,c) for a≠b≠c.
   • S is not antisymmetric because there are pairs (a,b) and (b,a) for a≠b.

3. Relation T={(1,2),(2,3),(1,3), (4,4)}:

   • T is not reflexive because not every element in set A is related to itself.
   • T is not symmetric because for every pair (a,b), there is not always a pair (b,a).
   • T is transitive because for every pair (a,b) and (b,c), there is a pair (a,c).
   • T is not antisymmetric because there are no pairs (a,b) and (b,a) for a≠b.

4. Relation U={(1,3),(2,4),(3,1),(4,2),(1,1),(1,4)}:

   • U is not reflexive because not every element in set A is related to itself.
   • U is not symmetric because for every pair (a,b), there is not always a pair (b,a).
   • U is not transitive because there are no pairs (a,b), (b,c) and (a,c) for a≠b≠c.
   • U is not antisymmetric because there are pairs (a,b) and (b,a) for a≠b.