For any partial order there may be a number of linear orders that are consistent. For example, for the partial order (P (A), ⊂) with A = {1, 2, 3}, the ordering ∅ ≺ {1} ≺ {2} ≺ {1, 2} ≺ {3} ≺ {1, 3} ≺ {2, 3} ≺ {1, 2, 3} does not violate the partial order, i.e. if a ≺ b → b ̸ ⊆ a. Find all the consistent linear orders for (P (A), ⊂) when A = {1, 2, 3, 4}.
For any partial order there may be a number of linear orders that are consistent. For example, for the partial order (P (A), ⊂) with A = {1, 2, 3}, the ordering ∅ ≺ {1} ≺ {2} ≺ {1, 2} ≺ {3} ≺ {1, 3} ≺ {2, 3} ≺ {1, 2, 3} does not violate the partial order, i.e. if a ≺ b → b ̸ ⊆ a. Find all the consistent linear orders for (P (A), ⊂) when A = {1, 2, 3, 4}.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.3: Systems Of Inequalities
Problem 19E
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Question
For any partial order there may be a number of linear orders that are consistent. For example,
for the partial order (P (A), ⊂) with A = {1, 2, 3}, the ordering
∅ ≺ {1} ≺ {2} ≺ {1, 2} ≺ {3} ≺ {1, 3} ≺ {2, 3} ≺ {1, 2, 3}
does not violate the partial order, i.e. if a ≺ b → b ̸ ⊆ a. Find all the consistent linear orders for
(P (A), ⊂) when A = {1, 2, 3, 4}.
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