Final answer:
The question does not provide enough information to determine if the relation R is reflexive, symmetric, transitive, or an equivalence relation. The set [x] just defines a collection of elements in A related to x by R, but the properties of R must be known to categorize the relation.
Step-by-step explanation:
The set [x] = a is an example of a collection of all elements a in set A that are related to an element x by the relation R. This definition alone does not determine whether the relation R is reflexive, symmetric, transitive, or an equivalence relation. These properties must be proven through the characteristics of the relation R itself.
- A relation is reflexive if every element is related to itself.
- A relation is symmetric if whenever an element a is related to an element b, then b is also related to a.
- A relation is transitive if whenever an element a is related to an element b and b is related to an element c, then a is also related to c.
- An equivalence relation is a relation that is reflexive, symmetric, and transitive.
Without more information about the properties of R, we cannot conclude which of these characteristics it possesses.