23.2.5. Example with Positive Integers

Let me give you some more examples of partial ordering here. So, I consider the set of all positive integers, so this set ℤ₊ is the set of positive integers. And I defined a relation divides which I am denoting by |. So, my relation R is the divides (|) relationship. And I say that (a,b) ∈ R if a | b, and a,b ∈ ℤ₊. Otherwise, a is not related to b. So, now it is easy to see that this relationship is reflexive because every positive integer divides itself. This relationship is antisymmetric, because you cannot have two different positive integers simultaneously dividing each other if a divides b as well as b divides a then that is possible only when both the integers are same. And if a | b and b | c, then you have a | c. So, it satisfies the transitivity property. So, this is an example of a partial ordering.

In this chunk, the lecturer describes a specific example of a partial ordering using positive integers and the 'divides' relation.

1. Set of Positive Integers: The set of all positive integers is represented as ℤ₊.
2. Divides Relation: The divides relation, denoted by '|', indicates that one number can divide another without leaving a remainder. For instance, if a = 2 and b = 4, we say 2 divides 4, written as 2 | 4.
3. Reflexive Property: The divisibility relation is reflexive because any positive integer divides itself (e.g., 5 | 5).
4. Antisymmetric Property: This property states that if a divides b and b divides a, then a and b must be the same integer. Thus, two different integers can't divide each other.
5. Transitive Property: The transitive property states that if a | b and b | c, then it logically follows that a | c.
   Overall, these three properties fulfill the requirements for a partial ordering.

Think of a scenario where you're stacking books in a library. If each book represents a positive integer, and the concept of 'dividing' illustrates how many books can be stacked together without exceeding the maximum limit. For example, if you have 2 books that can each be part of a stack of 4 books, and 4 can be part of a stack of 8 books, you see the relation that allows you to build larger stacks sequentially, which is analogous to how divisibility works.

Now, let me define another relation here my relation here is ⊆. My R here is the subset relationship, which I am denoting by ⊆. And my elements are the elements of the power set of a set. So, the relation is not defined over the set S. I stress that the relation is defined over the power set of S. So, my elements are here subsets of S and I say that a subset A is related to the subset B, if A ⊆ B. That is my relation. Again this relation satisfies the reflexive property because A ⊆ A. It satisfies the antisymmetric property because you cannot have two different subsets A and B, A ⊆ B and B ⊆ A, because that is the case that means A = B. And it satisfies the requirement of a transitive relation. If A ⊆ B and B ⊆ C, then that means that A ⊆ C.

In this chunk, the lecturer gives another example of a partial ordering using the subset relation.

1. Subset Relation: The subset relation, represented as '⊆', indicates that all elements in set A are also contained in set B.
2. Power Set: The relation is defined over the power set of a set S, which is the set of all possible subsets of S.
3. Reflexive Property: Every set is a subset of itself, satisfying the reflexive property (A ⊆ A).
4. Antisymmetric Property: If A is a subset of B (A ⊆ B) and B is also a subset of A (B ⊆ A), then A and B must be the same set. Therefore, two distinct sets cannot share this property.
5. Transitive Property: The transitive property applies if A ⊆ B and B ⊆ C, leading to the conclusion that A ⊆ C.
   These properties collectively qualify the subset relationship as a partial ordering.

Consider a menu of food items at a restaurant. Each menu represents a set of dishes, while the different categories of dishes (like appetizers, main courses) represent subsets. If you think of a large menu as a full set, each category can be seen as a subset of dishes within the larger menu. Just as every category can exist on its own (reflexive), if an appetizer category and a main-course category were to match, they must essentially contain the same dishes (antisymmetric), and if one category can contain dishes found in another, it leads to a clear hierarchy of categories (transitive).

Similarly, if I take the set of integers, ℤ, and if my relationship is the less than equal to relationship where integer a is related to integer b provided a ≤ b. Then again this satisfies the reflexive property, antisymmetric property and transitive property and hence this is an example of a partial ordering.

In this chunk, the lecturer describes the less than or equal to relationship with integers as another example of partial ordering.

1. Set of Integers: The set of integers is denoted as ℤ.
2. Less Than or Equal Relation: The relationship is defined where an integer a is related to an integer b if a ≤ b.
3. Reflexive Property: This holds true since any integer is always less than or equal to itself (e.g., 3 ≤ 3).
4. Antisymmetric Property: If a ≤ b and b ≤ a, this implies that a and b must be the same integer.
5. Transitive Property: If a ≤ b and b ≤ c, then it follows that a ≤ c.
   This relationship demonstrates the key properties of a partial ordering.

Imagine a race where several runners are competing. Each runner represents an integer, and their finishing positions can be related through this 'less than or equal' relationship. If one runner finishes before another (meaning they placed higher), you can say one has a lesser or equal position compared to the other. The reflexive property is demonstrated as every runner is in their respective spot (themselves), and the relationships between runners can be used to determine who qualified for the next round (transitive).

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