43 - In the definition of partial order, notice that...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: In the definition of partial order, notice that anti-symmetric is not the opposite of symmetric, since a relation can be both symmetric and anti-symmetric: 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 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 : • 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 non-empty 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 ...
View Full Document

Ask a homework question - tutors are online