91Ó°ÊÓ

The intersection graph of a collection of sets \(A_{1}\) , \(A_{2}, \ldots, A_{n}\) is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph of these collections of sets. a) \(A_{1}=\\{0,2,4,6,8\\}, A_{2}=\\{0,1,2,3,4\\}\) \(A_{3}=\\{1,3,5,7,9\\}, A_{4}=\\{5,6,7,8,9\\}\) \(A_{5}=\\{0,1,8,9\\}\) b) \(A_{1}=\\{\ldots,-4,-3,-2,-1,0\\}\) \(A_{2}=\\{\ldots,-2,-1,0,1,2, \ldots\\}\) \(A_{3}=\\{\ldots,-6,-4,-2,0,2,4,6, \ldots\\}\) \(A_{4}=\\{\ldots,-5,-3,-1,1,3,5, \ldots\\}\) \(A_{5}=\\{\ldots,-6,-3,0,3,6, \ldots\\}\) c) \(A_{1}=\\{x | x < 0\\}\) \(A_{2}=\\{x |-1 < x < 0\\}\) \(A_{3}=\\{x | 0 < x < 1\\}\) \(A_{4}=\\{x |-1 < x < 1\\}\) \(A_{5}=\\{x | x > -1\\}\) \(A_{6}=\mathbf{R}\)

Step by step solution

01

- Identify the vertices

Each set in the collection corresponds to a vertex in the intersection graph. For example, in part (a), the sets are \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\), and \(A_{5}\), and therefore, the vertices of the graph are labeled as 1, 2, 3, 4, and 5 respectively.

02

- Determine the intersections for part (a)

Examine the intersections between each pair of sets:- \(A_{1} \cap A_{2} = \{0, 2, 4\}\) (non-empty)- \(A_{1} \cap A_{3} = \varnothing\) (empty)- \(A_{1} \cap A_{4} = \{6, 8\}\) (non-empty)- \(A_{1} \cap A_{5} = \{0\}\) (non-empty)- \(A_{2} \cap A_{3} = \{1, 3\}\) (non-empty)- \(A_{2} \cap A_{4} = \varnothing\) (empty)- \(A_{2} \cap A_{5} = \{0, 1\}\) (non-empty)- \(A_{3} \cap A_{4} = \{5, 7, 9\}\) (non-empty)- \(A_{3} \cap A_{5} = \{1, 9\}\) (non-empty)- \(A_{4} \cap A_{5} = \{8, 9\}\) (non-empty)

03

- Construct the intersection graph for part (a)

Draw vertices labeled 1 through 5. Draw edges between vertices based on the intersections identified:- Edge between 1 and 2- Edge between 1 and 4- Edge between 1 and 5- Edge between 2 and 3- Edge between 2 and 5- Edge between 3 and 4- Edge between 3 and 5- Edge between 4 and 5

04

- Determine the intersections for part (b)

Examine the intersections between each pair of sets:- \(A_{1} \cap A_{2} = \{0, -1, -2, \text{etc.}\}\) (non-empty)- \(A_{1} \cap A_{3} = \{0, -2, -4, \text{etc.}\}\) (non-empty)- \(A_{1} \cap A_{4} = \{0, -1, \text{etc.}\}\) (non-empty)- \(A_{1} \cap A_{5} = \{0, -3, -6, \text{etc.}\}\) (non-empty)- \(A_{2} \cap A_{3} = \{0, 2, \text{etc.}\}\) (non-empty)- \(A_{2} \cap A_{4} = \{1, -1, \text{etc.}\}\) (non-empty)- \(A_{2} \cap A_{5} = \{3, 0, -3, \text{etc.}\}\) (non-empty)- \(A_{3} \cap A_{4} = \{1, -1, \text{etc.}\}\) (non-empty)- \(A_{3} \cap A_{5} = \{3, 0, -3, \text{etc.}\}\) (non-empty)- \(A_{4} \cap A_{5} = \{3, -3, \text{etc.}\}\) (non-empty)

05

- Construct the intersection graph for part (b)

Draw vertices labeled 1 through 5. Draw edges between every pair of vertices because all sets intersect with each other.

06

- Determine the intersections for part (c)

Examine the intersections between each pair of sets:- \(A_{1} \cap A_{2} = A_{2} \) (non-empty)- \(A_{1} \cap A_{3} = \varnothing \) (empty)- \(A_{1} \cap A_{4} = A_{2} \) (non-empty)- \(A_{1} \cap A_{5} = A_{2} \) (non-empty)- \(A_{1} \cap A_{6} = A_{1} \) (non-empty)- \(A_{2} \cap A_{3} = \varnothing \) (empty)- \(A_{2} \cap A_{4} = A_{2} \) (non-empty)- \(A_{2} \cap A_{5} = A_{2} \) (non-empty)- \(A_{2} \cap A_{6} = A_{2} \) (non-empty)- \(A_{3} \cap A_{4} = A_{3} \) (non-empty)- \(A_{3} \cap A_{5} = A_{3} \) (non-empty)- \(A_{3} \cap A_{6} = A_{3} \) (non-empty)- \(A_{4} \cap A_{5} = A_{4} \) (non-empty)- \(A_{4} \cap A_{6} = A_{4} \) (non-empty)- \(A_{5} \cap A_{6} = A_{5} \) (non-empty)

07

- Construct the intersection graph for part (c)

Draw vertices labeled 1 through 6. Draw edges for every pair of intersecting sets based on the intersections found: - Edge between 1 and 2- Edge between 1 and 4- Edge between 1 and 5- Edge between 1 and 6- Edge between 2 and 4- Edge between 2 and 5- Edge between 2 and 6- Edge between 3 and 4- Edge between 3 and 5- Edge between 3 and 6- Edge between 4 and 5- Edge between 4 and 6- Edge between 5 and 6

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

set theory

Set theory deals with the concept of sets, which are collections of objects. Understanding set theory is crucial when constructing intersection graphs.
In the given problem, each set contains numbers and is represented by a vertex. Sets can intersect, meaning they share common elements. When two sets have a non-empty intersection, we draw an edge connecting their respective vertices in the graph.
Here are some key terms in set theory relevant to this problem:

• Element: An object within a set.
• Intersection: The set containing all elements common to both sets. For example, if set A = {1, 2, 3} and set B = {2, 3, 4}, their intersection A ∩ B = {2, 3}.
• Empty set: A set with no elements, denoted as ∅. If two sets do not share any elements, their intersection is empty.

To determine intersections, examine pairs of sets and list their common elements.
If there's at least one common element, the intersection is non-empty, and we create a corresponding edge in the graph.

graph theory

Graph theory involves the study of graphs, which consist of vertices (nodes) and edges (lines connecting nodes).
In the context of this problem, an intersection graph is created based on the intersections of sets.
Each vertex represents a set, and an edge is drawn between vertices if their corresponding sets intersect.
Steps to construct the intersection graph:

• Identify vertices: Each set in the collection is a vertex in the graph. For example, sets A1, A2, A3 correspond to vertices 1, 2, 3.
• Determine intersections: Check pairs of sets for common elements. If a pair of sets has a non-empty intersection, an edge is added between their vertices.
• Draw the graph: Start with the identified vertices and add edges based on the determined intersections.

Graph theory provides a visual and structured way to represent relationships, making it easier to analyze connections between sets.
By understanding this, you can build accurate intersection graphs.

discrete mathematics

Discrete mathematics studies structures that are fundamentally discrete rather than continuous.
It includes topics like set theory, graph theory, and combinatorics.
In this problem, everything is discrete: sets contain finite elements, intersections are explicitly determined, and the resulting graphs are finite.
To solve the intersection graph problem using discrete mathematics principles:

• Understand the nature of sets: Each set is a collection of distinct numbers.
• Apply intersection operations: Use set theory to find common elements between each pair of sets.
• Construct the graph: Use graph theory principles to represent these intersections visually.

Discrete mathematics helps provide a clear framework for problem-solving.
By studying discrete structures, you can understand complex relationships and connections between finite elements, enhancing your ability to tackle intersection graph problems.