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# Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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Homework 14 - MATH 310 0101 - Summer I 2010 - Due by Wednesday, June 30 1. Exercise 7.6 2. Exercise 7.8 3. Let A and B be partial orders on A and B respectively. Deﬁne a relation on A × B by ( a,b ) ( a 0 ,b 0 ) if and only if either a A a 0 and a 6 = a 0 or a = a 0 and b B b 0 . Show that deﬁnes a partial order on A × B . (This is called the lexicographic or dictionary order on A × B .) Also show that if A and B are total orderings then so is . 4. Let A and B be partial orderings on A and B respectively. We call f : A B an order isomorphism if f is a bijection and for every x,y A , x A y if and only if f ( x ) B f ( y ). Show that the following functions are order isomorphisms. (a) f : N S deﬁned by n 7→ n !, where S is the set { n ! : n N } and the orders on N and S are given by the usual order on natural numbers. (b) f : N S deﬁned by n 7→ [ n ] where S = { [ n ] : n N }
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Unformatted text preview: , the order on N is the usual order, and the order on S is ⊂ . 5. A relation ≺ on a set S is called irreﬂexive if for every x ∈ S , x 6≺ x . We call a relation a strict partial order if it is irreﬂexive and transitive. Prove that a strict partial order is antisymmetric. 6. Let C [0 , 1] be the set of continuous real-valued functions on [0 , 1]. De-ﬁne a relation ≺ on C [0 , 1] by f ≺ g if and only if Z 1 ( g-f )( x ) dx > . Show that ≺ is a strict partial order. 7. Let ≺ be a strict partial order on S . Deﬁne a relation ± on S by x ± y if and only if x ≺ y or x = y . Show that ± is a partial order on S....
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