Question

What is the minimal element of the set of all subsets of a given set \(X\), partially ordered by set inclusion. What is the maximal element?

Short answer

The minimal element is the empty set \( \emptyset \), and the maximal element is the set \( X \) itself.

Step-by-step solution

Understand the Set Inclusion Order

The problem asks us to find a minimal and maximal element of the power set of some set \( X \), with partial ordering defined by set inclusion. Essentially, we're looking at all possible subsets of \( X \) and arranging them in a way that subset \( A \) is considered less than or equal to subset \( B \) if \( A \) is contained within \( B \).

Identify the Minimal Element

In a partially ordered set of subsets, the minimal element is the one that is not a superset of any other subset except itself. The empty set \( \emptyset \) is the subset contained in every other subset of \( X \). Therefore, \( \emptyset \) is the minimal element of the power set.

Determine the Maximal Element

To find the maximal element, we look for the subset that does not have any subset (except itself) contained in it. The maximal element in the set inclusion order is the set \( X \) itself, as it is not a proper subset of any larger set within the power set. Every other subset of \( X \) is contained in \( X \), making \( X \) the maximal element.

Key concepts

Power Set

The power set of a given set \( X \) is an important concept in set theory. It is the set of all possible subsets that can be formed from \( X \). This includes both the empty set \( \emptyset \) and \( X \) itself. For example, if \( X = \{a, b\} \), then the power set of \( X \), denoted as \( P(X) \), would be \( \{ \emptyset, \{a\}, \{b\}, \{a, b\} \} \). Thus, the power set contains every combination of the elements of \( X \).
Understanding power sets is crucial, as they help explore the possible relationships between subsets and provide a foundation for understanding concepts like partial order and set inclusion.

Partial Order

A partial order is a way to arrange elements of a set such that some pairs of elements are comparable in a transitive, asymmetric, and reflexive way. In terms of subsets, partial order is established by the concept of set inclusion. We can compare subsets based on whether one subset is contained within another. This relationship between subsets respects elements' hierarchies, enabling us to define what it means for a subset to be less than, equal to, or greater than another subset in terms of inclusion. This order is not total, meaning not all pairs of subsets will necessarily be comparable.

Set Inclusion

Set inclusion is a fundamental concept where a set \( A \) is considered to be a subset of set \( B \) if every element of \( A \) is also an element of \( B \). This is denoted as \( A \subseteq B \). For example, if \( A = \{1, 2\} \) and \( B = \{1, 2, 3\} \), then \( A \subseteq B \).
Set inclusion naturally creates a hierarchy among the elements in a power set, as smaller subsets are included in larger ones. This hierarchy forms the basis for establishing partial order among sets.

Minimal Element

In the context of sets, the minimal element of a power set is the smallest element with respect to set inclusion. Since set inclusion orders subsets by containment, the minimal element is the subset that is contained in every other subset. In any power set, the minimal element is the empty set \( \emptyset \).
The empty set is unique because it has no elements and is a subset of every other set, meaning it holds the smallest possible position within the order defined by set inclusion.

Maximal Element

Conversely, the maximal element in a power set is the largest subset within the context of set inclusion. This means no other subset can contain it apart from itself. In the power set of a set \( X \), the set \( X \) itself is the maximal element. This is because \( X \) encompasses all the elements under consideration and thus cannot be contained within any larger subset of the power set.

• The set \( X \) serves as an upper bound for inclusion in the hierarchy of subsets and holds a significant position as every other subset is contained within it.