This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1. (a) Let ( A; & ) be a wellordered set, B ¡ A and f : A onto ! B be an order isomorphism of ( A; & ) onto ( B; & ) . Prove that for every a 2 A , a & f ( a ) . (b) Let ( A; & ) be a wellordered set and a 2 A . Recall that the set I a = f x 2 A j x ¢ a g is called an initial segment of A . Use part (a) to prove that ( A; & ) cannot be order isomorphic to one of its initial segments. 2. An ideal in a partial ordered set ( X; & ) is a set I ¡ X such that ( x 2 I ) ^ [( y 2 X ) ^ ( y & x )] ) y 2 I . Prove that every ideal I in a wellordered set ( X; & ) is either an initial segment of ( X; & ) or X . 3. Let ( X; & ) be a wellordered set and B be a nonempty subset of X . Then [ x 2 B I x is either equal to an initial segment in ( X; & ) or X (here I x is the initial segment in ( X; & ) ). 4. Prove that any set of ordinal numbers is a partially ordered set w.r.t. to the standard order & £ ¡for ordinals....
View
Full Document
 Spring '08
 Ruan
 Addition, Natural number, initial segment, A1. Commutative law

Click to edit the document details