# Answer on Abstract Algebra Question for John

Question #9298

0. Show that there exist infinitely many finite sets that are not transitive and infinitely man infinite sets that are not transitive.

1. Show: A set X is transitive if and only if X is a subset of P(X).

2. Show: if every X *belongs to* S is transitive, then union(S) is transitive.

3. Show: an ordinal α is a natural number if and only if every nonempty subset of α has a greatest element.

4.Show: If a set of ordinals X does not have a greatest element, then sup X is a limit ordinal.

5. Show: If X is a nonempty set of ordinals, then intersection(X) is an ordinal. Moreover, intersection(X) is the least element of X.

1. Show: A set X is transitive if and only if X is a subset of P(X).

2. Show: if every X *belongs to* S is transitive, then union(S) is transitive.

3. Show: an ordinal α is a natural number if and only if every nonempty subset of α has a greatest element.

4.Show: If a set of ordinals X does not have a greatest element, then sup X is a limit ordinal.

5. Show: If X is a nonempty set of ordinals, then intersection(X) is an ordinal. Moreover, intersection(X) is the least element of X.

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