174k views
4 votes
Show that any element of a hereditarily transitive set is hereditarily transitive.

a) Write a proof showing that if x is in a hereditarily transitive set, then every element of x is also in a hereditarily transitive set.
b) Explain the concept of hereditary transitivity without proving the statement.
c) Provide an example of a set that is hereditarily transitive.
d) Disprove the statement by giving a counterexample.

1 Answer

1 vote

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.

User Vinnydiehl
by
8.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.