Question

(a) Explain why the relation "is older than or the same age" is a partial order. (b) Explain why the relation "is older than" is not a linear order.

Short answer

(a) "Is older than or the same age" is reflexive, antisymmetric, and transitive, making it a partial order. (b) "Is older than" lacks comparability, so it's not a linear order.

Step-by-step solution

Define a Partial Order

A relation is a partial order if it is reflexive, antisymmetric, and transitive. Let's break this down for the relation "is older than or the same age."

Check Reflexivity

The relation "is older than or the same age" is reflexive because every person is at the same age as themselves. Therefore, for any person \(a\), \(a\) is related to \(a\).

Check Antisymmetry

The relation is antisymmetric because if person \(a\) is older than or the same age as person \(b\), and \(b\) is older than or the same age as \(a\), then \(a\) and \(b\) must be the same age. This implies \(a = b\).

Check Transitivity

The relation is transitive because if person \(a\) is older than or the same age as person \(b\), and \(b\) is older than or the same age as person \(c\), then \(a\) is older than or the same age as \(c\).

Conclusion on Partial Order

Since the relation "is older than or the same age" satisfies reflexivity, antisymmetry, and transitivity, it is a partial order.

Define a Linear Order

A relation is a linear order if it is a partial order and comparable, meaning for any two elements \(a\) and \(b\), either \(a\) is related to \(b\) or \(b\) is related to \(a\).

Comparability Test for "Is Older Than"

The relation "is older than" is not comparable because there can be cases where neither \(a\) is older than \(b\) nor \(b\) is older than \(a\), specifically when \(a\) and \(b\) are the same age. Hence, it's not a linear order.

Key concepts

Reflexivity

Reflexivity is an essential property of a partial order that implies each element relates to itself within the given relation. Let's consider the relation "is older than or the same age." When we say this relation is reflexive, we mean that for any person, say person \( a \), this person is always as old as themselves.
Therefore, for reflexivity, every time we pick a person, the statement "person \( a \) is older than or the same age as person \( a \)" holds true. This self-relating aspect is what makes the relation reflexive, which is a requirement for being a partial order.

• Reflexivity ensures that each element is in the relationship with itself.
• For any element \( a \), you must be able to say \( a \sim a \).
• In our age example, every person is as old as themselves, thus satisfying reflexivity.

As a result, reflexivity helps establish a foundational relationship in sets where each element has a basic point of reference to itself.

Antisymmetry

Antisymmetry is a crucial property that sets boundaries within a relation to ensure uniqueness. It implies that if one element is related to another, and vice versa, then they must be identical. Let's delve into how this applies to the relation "is older than or the same age."
In terms of age, if person \( a \) is older than or the same age as person \( b \), and person \( b \) is older than or the same age as person \( a \), the only conclusion is that they must be of the same age. It insists on non-distinct elements being equal.

• Antisymmetry posits \( a \sim b \) and \( b \sim a \) means \( a = b \).
• In age terms, this means if person \( a \) and person \( b \) share this bi-directional relation, they are identical in age.
• It differentiates the concept from symmetrical relationships.

Antisymmetry provides a necessary check that keeps the structure of relations non-redundant, and in the context of partial orders, maintains the integrity of the order itself.

Transitivity

Transitivity is a property ensuring that a relation logically progresses across elements. If one element relates to a second, and the second relates to a third, then the first must relate to the third, as demonstrated in the relation "is older than or the same age."
When applying transitivity to age, if person \( a \) is older than or the same age as person \( b \), and person \( b \) is older than or the same age as person \( c \), then it follows that person \( a \) is older than or the same age as person \( c \). The concept dictates coherence in the way elements connect sequentially.

• Transitivity posits \( a \sim b \) and \( b \sim c \) result in \( a \sim c \).
• It creates a chain of relationships ensuring consistency throughout the set.

Transitivity solidifies connections between more than two elements, maintaining an organized flow within the order, which is vital for the definition of partial orders.