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Unformatted text preview: In the deﬁnition of partial order, notice that antisymmetric is not the opposite of symmetric, since a relation can be both symmetric and antisymmetric: for example, the identity relation or the empty relation. EXAMPLE 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 preorder. 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 : • lexicographic order: (a1 , b1 ) ≤L (a2 , b2 ) iff (a1 <A a2 ) ∨ (a1 = a2 ∧ b1 ≤B b2 ). If (A, ≤) and (B , ≤) are both total orders, then the lexicographic order on A × B will be total. By contrast, the product order will in general only be partial. For any partially ordered sets (A, ≤) and (B , ≤), the product order is contained in the lexicographic order. 7. (For interest, gives ordering for words in a dictionary) For any totally ordered (ﬁnite) alphabet A, the sets A∗ = { } ∪ A ∪ A2 ∪ A3 ∪ . . . is the set if all strings made from that alphabet, with denoting the empty string. The full lexicographic order ≤ F on A∗ is deﬁned as follows. Given two words u, v ∈ A∗ , if u = then u ≤F v and if v = then v ≤F u. Otherwise, both u and v are nonempty so we can write u = u1 x and v = v1 y where u1 and v1 are the ﬁrst letters of u and v respectively. Now u ≤F v ⇔ (u = ) ∨ (u1 <A v1 ) ∨ (u1 = v1 ∧ x ≤F y ). • product order: (a1 , b1 ) ≤P (a2 , b2 ) iff (a1 ≤A a2 ) ∧ (b1 ≤B b2 ) 43 ...
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 Spring '09
 Koskesh
 Math

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