Question

Represent by a digraph the partial order defined on \(P(\\{1,2,3,4\\})\) where the relation is set inclusion.

Short answer

Draw a digraph showing set inclusion among the subsets of \( \{1, 2, 3, 4\} \) forming a Hasse diagram.

Step-by-step solution

Understand the Power Set

First, we need to understand what the power set, denoted by \( P \), of a set is. The power set of \( \{1, 2, 3, 4\} \), is the set of all possible subsets of \( \{1, 2, 3, 4\} \). This includes the empty set \( \emptyset \), and every combination of elements up to the set itself. The power set of a set with \( n \) elements has \( 2^n \) elements. Hence, there are \( 2^4 = 16 \) subsets in \( P(\{1, 2, 3, 4\}) \).

List the Subsets

List all the subsets of \( \{1, 2, 3, 4\} \): \( \emptyset, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}, \{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}, \{2, 3, 4\}, \{1, 2, 3, 4\} \).

Define the Relation

The partial order relation given is set inclusion. Denote it by \( \subseteq \). This means that if subset \( A \) is contained within subset \( B \), we draw an arrow from \( A \) to \( B \) in the digraph. This relation is reflexive, antisymmetric, and transitive.

Identify Comparable Pairs

Within the power set, identify the pairs of sets where one is a subset of the other. For example, \( \emptyset \subseteq \{1\} \), \( \{1\} \subseteq \{1, 2\} \), etc. Each subset has arrows pointing to the sets that it can be contained in.

Draw the Digraph

Begin drawing the digraph starting with the simplest set, usually the empty set \( \emptyset \), and show its relation to all the subsets. Continue with single-element subsets relating to double-element subsets, and so on, until you reach the full set \( \{1, 2, 3, 4\} \). Each arrow represents the inclusion relation.

Key concepts

Power Set

A power set is a fundamental concept in set theory. It refers to the collection of all possible subsets of a given set. For example, consider the set \( \{1, 2, 3, 4\} \). The power set includes: - The empty set \( \emptyset \), which contains no elements. - All individual singleton sets, such as \( \{1\}, \{2\}, \{3\}, \{4\} \). - Pairs like \( \{1, 2\}, \{2, 3\} \), and so forth. - Triplets like \( \{1, 2, 3\} \), etc. - The set itself \( \{1, 2, 3, 4\} \).
The number of subsets in a power set is determined by the formula \( 2^n \), where \( n \) is the number of elements in the original set. Thus, for a set with 4 elements, the power set has \( 2^4 = 16 \) subsets. Understanding the power set helps create a deeper insight into organizing and analyzing the relationships between different sets.

Set Inclusion

Set inclusion is a significant type of relationship between two sets. It is signified by the symbol \( \subseteq \). This notation signifies that one set, say \( A \), is a subset of another, \( B \), if every element of \( A \) is also present in \( B \).
This relation is consistent, structured, and follows:- **Reflexivity:** Any set \( A \) is always a subset of itself, \( A \subseteq A \).- **Antisymmetry:** If \( A \subseteq B \) and \( B \subseteq A \), then \( A = B \).- **Transitivity:** If \( A \subseteq B \) and \( B \subseteq C \), then \( A \subseteq C \).
Set inclusion is crucial because it establishes a partial order among elements of a power set. It helps to determine the hierarchical structure in sets, which is critical in mathematics and computer science.

Directed Graphs (Digraphs)

Directed graphs, often called digraphs, are graphical representations of relationships with direction. In the context of partial orders, digraphs visually depict the set inclusion relation.
Each element or subset is represented by a node. An arrow or directed edge from one node to another indicates a relationship of inclusion. For example, if there is an arrow from node \( A \) to node \( B \), it represents that set \( A \) is included in set \( B \).
Key characteristics of digraphs include:- **Directionality:** The relationships or edges have a definite orientation.- **No Cycles:** In a partial order, there should be no cycles, meaning no set can directly or indirectly include itself in a loop.
Drawing a digraph involves careful analysis of set inclusions, beginning with the smallest elements (often the empty set) and mapping out all relationships until the largest set is reached. This visualization makes it easier to comprehend complex relations between elements and assess their partial ordering.