Partial order example

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An AI-generated summary of this video lecture.

This lecture introduces partial ordering sets (Posets) and demonstrates how to construct the relation for a specific example. The instructor begins by defining a Poset as a set A with a partial order relation R, denoted [A, R]. Examples of Posets include [A, /] for divisibility, [A, <=] for less-than-or-equal-to, and [P(S), ⊆] for subset relations. The main worked example uses the set A = {1, 2, 3, 6} with the divisibility relation. The instructor defines (a,b) ∈ R if a divides b, which is equivalent to b/a being an integer in A. The instructor then explicitly lists the ordered pairs that satisfy this condition: (1,2), (1,3), (1,6), (2,6), (3,6), and (6,6). These pairs are circled to highlight the complete relation. The lecture focuses on translating an abstract definition into a concrete set of ordered pairs, showing how the divisibility rule generates each pair in the relation.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The slide titled 'Partial ordering set (Poset)' defines a Poset as a set A with a partial order relation R, denoted [A, R]. Examples listed include [A, /], [A, <=], and [P(S), ⊆]. The instructor selects the divisibility example and writes A = {1, 2, 3, 6} in green chalk. The condition (a,b) ∈ R is defined as b/a ∈ A, meaning a divides b. The instructor begins listing ordered pairs such as (1,2), (1,3), and (1,6) to show how the relation is constructed from the set elements.

2. 2:00 – 3:33 02:00-03:33

   The instructor completes the list of ordered pairs for the divisibility relation on A = {1, 2, 3, 6}, adding (2,6), (3,6), and (6,6). The full set of pairs is enclosed in a large green oval to emphasize the complete relation. A fraction 24/12 appears on the right side of the board, possibly as an additional illustration. The lecture concludes by showing how the abstract definition (a,b) ∈ R if a divides b translates into a concrete set of ordered pairs, reinforcing the connection between the rule and its explicit enumeration.

The lecture teaches how to construct a partial ordering set (Poset) by defining the relation explicitly. The key concept is that a Poset [A, R] consists of a set A and a relation R that satisfies partial order properties. The worked example uses divisibility on {1, 2, 3, 6}, where (a,b) ∈ R means a divides b. The instructor demonstrates the method by checking each pair of elements to see if one divides the other, producing the relation {(1,2), (1,3), (1,6), (2,6), (3,6), (6,6)}. This approach helps students understand how to move from an abstract definition to a concrete representation of the relation. The circling of the pairs emphasizes that this complete set constitutes the relation R for the given Poset.