540hw3 - 1. (a) Let ( A; & ) be a well-ordered...

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Unformatted text preview: 1. (a) Let ( A; & ) be a well-ordered 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 well-ordered 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 well-ordered set ( X; & ) is either an initial segment of ( X; & ) or X . 3. Let ( X; & ) be a well-ordered set and B be a non-empty 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....
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540hw3 - 1. (a) Let ( A; & ) be a well-ordered...

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