# ws805 - (b) Is there a least element? A greatest element?...

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Rob Bayer Math 55 Worksheet August 5, 2009 Partial Orders 1. Determine whether each of the following are partial orders. For those that are, decide whether or not they are linear orders. (a) ( N , ) (b) (“all people”, “is an ancestor of”) (Suppose you are considered an ancestor of yourself) (c) ( { f | f : R R } ,f g ⇔ ∀ x f ( x ) < g ( x )) (d) ( Z , | ) (e) ( P ( { a,b,c } ) , ) 2. Draw the Hasse diagram for the partial order given by divisibility on the set { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } . What are the maximal and minimal elements? 3. Suppose ( P 1 , ± 1 ) and ( P 2 , ± 2 ) are partial orders. (a) Show that the lexicographic order given by ( a,b ) ± ( c,d ) ( a 1 c ( a = c b ± 2 d )) is a partial order on P 1 × P 2 (b) Show that the product order given by ( a,b ) ( c,d ) a 1 c b 2 d is also a partial order 4. Consider the poset ( { 2 , 4 , 6 , 9 , 12 , 18 , 27 , 36 , 48 , 60 , 72 } , |} ) (a) What are the maximal elements? What are the minimal elements?
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Unformatted text preview: (b) Is there a least element? A greatest element? (c) Find a lower bound for 60 , 72. (d) Find all upper bounds for 2 , 9 (e) Is this a lattice? 5. A poset is called well-founded if there is no inﬁnite descending chain of elements. A poset is called well-ordered if every non-empty subset has a least element. Consider the poset Z , ± given by x ≺ y iﬀ | x | < | y | (a) Show that this is a poset (b) Show that it is well-founded, but not well-ordered. 6. Show that the lexicographic order on the product of two well-founded posets is well-founded. 7. Show that the lexicographic order on the product of two well-ordered posets is a well-ordering....
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## This note was uploaded on 09/10/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at Berkeley.

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