Chapter 8

(i) Suppose that \(U\) is a non-empty set, and that \(P(U)\) is the set of subsets of \(U\). Show that if \(V \subseteq U\) and \(f: V \rightarrow P(U)\) is a mapping, then \(f\) is not onto. (Consider \(\\{x: x \in V, x \notin f(x)\\} .)\) (ii) Suppose that \(U\) is a non-empty set and that \(V \subseteq W \subseteq U\).Show that if \(f: V \rightarrow P(U)\) is one-one then there exists a one-one map \(g: W \rightarrow P(U)\) such that \(\left.g\right|_{V}=f\). (Use Zorn's lemma.)

Suppose that \(K(\alpha): K\) is a simple extension and that \(\alpha\) is transcendental over \(K\). Show that \(K(\alpha)\) is not algebraically closed.