Final answer:
The given relation is an equivalence relation since it satisfies reflexivity, symmetry, and transitivity. The equivalence classes for the set {1,2,3,4,5} are {[1,3], [2], [4], [5]} because each element is related to itself and the pairs (1,3) and (3,1) also relate 1 and 3 to each other.
Step-by-step explanation:
The question asks to determine whether a given relation is an equivalence relation on the set {1,2,3,4,5} and to list the equivalence classes if it is. An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. The given relation is given by the set of ordered pairs {(1,1),(2,2),(3,3),(4,4),(5,5),(1,3),(3,1)}.
Since the relation includes (1,1), (2,2), (3,3), (4,4), (5,5), it satisfies reflexivity because every element is related to itself. It satisfies symmetry because for every (x,y) in the relation, the pair (y,x) is also included, as we can see with the pairs (1,3) and (3,1). However, the relation does not satisfy transitivity: since (1,3) and (3,1) are in the relation, if the relation were transitive, then (1,1) would have to be in the relation as well. But (1,1) is already in the relation, and no additional information is provided to evaluate transitivity. Transitivity must be confirmed for all applicable cases, and there are no elements in the given relation that would violate transitivity.
Since the relation satisfies reflexivity, symmetry, and appears to satisfy transitivity, we can confirm that it is indeed an equivalence relation. The equivalence classes are the sets of elements that are related to each other. In this case, the equivalence classes are: {[1,3], [2], [4], [5]}. The equivalence class [1,3] contains both 1 and 3 since they are related, and the rest are single-element classes because there are no other relations specified between different elements.