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43 - In the denition of partial order notice that...

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In the definition of partial order, notice that anti-symmetric is not the oppo- site of symmetric, since a relation can be both symmetric and anti-symmetric: for example, the identity relation or the empty relation. E XAMPLE 5.2 1. The numerical orders on N , Z and R are total orders. The orders < are strict partial orders. 2. Division on N \{ 0 } is a partial order: n, m N . n m iff n divides m . 3. For any set A , the power set of A ordered by subset inclusion is a partial order. 4. Suppose ( A, A ) is a partial order and B A . Then ( B, B ) is a partial order, where B denotes the restriction of A to the set B . 5. Define a relation on formulae by: A B if and only if A B . Then is a pre-order. For example, false A true and A A B . 6. For any two partially ordered sets ( A, A ) and ( B, B ) , there are two important orders on the product set A × B : product order: ( a 1 , b 1 ) P ( a 2 , b 2 ) iff ( a 1 A a 2 ) ( b 1 B b 2 ) lexicographic order: ( a 1 , b 1 ) L ( a 2 , b 2 ) iff ( a 1 < A a 2 ) ( a 1 = a 2 b 1 B b 2 ) . If ( A, ) and ( B, ) are both total orders, then the lexicographic order
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