23.2.9. Comparable and Incomparable Elements

So, imagine you are given an arbitrary poset that so this less than equal to is an arbitrary relation R, which is a reflexive, antisymmetric and transitive. Now you take any two elements from the set S. They will be called comparable if a ≤ b or b ≤ a, incomparable otherwise. So, again to demonstrate these two concepts let us consider the divide relationship, that means you are less than equal to relationship is the divides relationship.

In a poset (partially ordered set), we can compare elements based on the defined relation. Elements ‘a’ and ‘b’ are considered comparable if one can be ordered before the other as per the relation R. If either 'a' is less than or equal to 'b' (written as 'a ≤ b') or 'b' is less than or equal to 'a' (written as 'b ≤ a'), they are comparable. On the other hand, if neither relation holds, they are called incomparable. For example, in the divide relationship, if '2 ≤ 4', it indicates that 2 divides 4, hence they are comparable. However, '2' does not divide '3', thus '2' and '3' are incomparable.

Think of comparable elements as friends who share common interests, like two friends who both love basketball. They can relate to each other based on their shared interest. On the other hand, imagine someone who loves hiking and another who loves video games. They are like incomparables; there’s no shared interest or relation that connects them based on their preferences.

So, that brings us to the definition of what we call as a total ordering. A total ordering is a special type of poset or partial ordering where you take any pair of elements from your set S, they will be comparable.

A total ordering is a specific kind of partial ordering where every pair of elements can be compared. That means, for any two elements 'a' and 'b' in the set, either 'a ≤ b' or 'b ≤ a' holds true. This is in contrast to general partial orderings, which can have elements that cannot be compared. The term 'total' indicates completeness in comparison among all elements.

Imagine a race with all participants. They can be placed in ranks, where one person finishes before another, leading all participants to have a position compared to one another (total ordering). In contrast, a group of people waiting for the bus might not have any defined order; some might know one another and can be compared, while others cannot, illustrating a partial ordering.

If I consider the less than equal to relationship namely, the numerical less than equal to relationship over the set of integers, then it is a total ordering. You take any pair of integers numerically either the first integer will be less than equal to the second integer or the second integer will be less than equal to the first integer.

The numerical less-than-or-equal-to relationship among integers serves as an example of total ordering. For instance, with integers like 3 and 5, we can say that '3 ≤ 5' is true because 3 is less than 5. This applies to any pair of integers, allowing for full comparability. On the other hand, the divides relationship (e.g., '2 divides 4') does not provide comparability for all integers, demonstrating a partial ordering.

Think of students in a class receiving grades. Every student can be compared to each other based on their scores. If one student scores higher, they are ranked above another. This is like a total order. In contrast, if you compare grades based on subjects, some students might excel in math while others in science, leading to situations where comparisons cannot be made between students based on their overall performance, resulting in a partial order.

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