Answer to Question #9298 in Abstract Algebra for John

Question

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.

Answer 1

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