Final answer:
For a set to be hereditarily transitive, every element within it must be a transitive set. Since every element of a hereditarily transitive set is transitive, and every element within those elements is also transitive, it shows that any element of a hereditarily transitive set is also hereditarily transitive. A counterexample to disprove the statement is not possible, as the statement is correct.
Step-by-step explanation:
Show that any element of a hereditarily transitive set is hereditarily transitive
For a set to be hereditarily transitive, every element of the set must themselves be transitive sets. A transitive set is a set in which every element that is a set is also a subset of the set. To show that any element of a hereditarily transitive set is also hereditarily transitive, consider any element x of a hereditarily transitive set H. Since H is hereditarily transitive, then x must be transitive. For any element y in x, since x is a subset of H, y is also an element of H. As H is hereditarily transitive, y must be transitive as well. Thus, every element of x is transitive, making x a hereditarily transitive set.
Explanation of hereditary transitivity
Hereditary transitivity means that not only is a set transitive, but every set within it is transitive, and this rule applies recursively down to all levels of sets within it.
Example of a set that is hereditarily transitive
An example of a hereditarily transitive set is the set {∅}. The empty set is transitive, and there are no further sets within it to consider, making it trivially hereditarily transitive.
Disproving the statement with a counterexample
statement given cannot be disproven by a counterexample, because it is a correct statement. All elements of a hereditarily transitive set are indeed hereditarily transitive by definition.