Total order relation

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The lecture introduces the concept of a total order relation in discrete mathematics. The instructor defines it as a relation where every pair of elements is comparable, meaning for all a and b in set A, either (a,b) or (b,a) belongs to the relation R. Using a dark slide, the instructor contrasts two examples: A = {1, 2, 3, 6} under the divisibility relation is not a total order because some elements are incomparable, while A = {1, 2, 4, 8} is a total order. The instructor then hand-draws a green trapezoid on the board listing ordered pairs such as (1,1), (1,2), (1,4), (1,8), (2,2), (2,4), and (2,8) to illustrate the comparable pairs in the total order case. The slide also references partial ordering sets (Posets) with examples like [A, /], [A, <=], and (P(S), ⊆].

Chapters

1. 0:00 – 1:31 00:00-01:31

   The instructor presents a slide titled 'Total order relation' with the definition: every pair of elements are comparable i.e. either (a, b) or (b, a) ∈ R for ∀ a, b ∈ A. The slide shows the example 'A = {1, 2, 3, 6}, then Poset [A,/] is not a total order relation' with the bracketed part circled in green, and 'but A = {1,2,4,8} will be' with the set underlined in green. The instructor points to these lines and later draws a green trapezoid on the board listing ordered pairs (1,1), (1,2), (1,4), (1,8), (2,2), (2,4), and (2,8) to demonstrate the comparable pairs. The slide also lists 'Partial ordering set (Poset): [A, /], [A, <=], (P(S), ⊆]' as related context.

The central idea is the definition of a total order relation: every pair of elements in the set must be comparable under the relation. The instructor uses two contrasting examples to illustrate this: {1, 2, 3, 6} under divisibility fails because 3 and 6 are not comparable in the required way (actually 3 divides 6, but 2 and 3 are incomparable), while {1, 2, 4, 8} succeeds because every pair is comparable under divisibility. The hand-drawn trapezoid with ordered pairs provides a concrete visualization of the relation's structure. The connection to Posets is noted, as a total order is a special case of a partial order where comparability holds for all pairs.