Question

The table below gives the names of airlines and several cities that each flies to from Chicago. The table also gives the number of miles for each flight. List all the triples \((X, Y, Z)\) of the temary relation defined by those triples for which airline \(X\) flies \(Y\) miles to city \(Z\). $$\begin{array}{|lr|ll||ll|}\hline \text { TWA } & & \text { Pan Am } & & \text { Piedmont } & \\\\\hline \text {Topeka } & 603 & \text { Bombay } & 7809 & \text { Peoria } & 170 \\\\\text { Kansas City } & 510 & \text { Seattle } & 2052 & \text { Albany } & 816 \\\\\text { Phoenix } & 1742 & \text { Anaheim } & 2025 & \text { Atlanta } & 717 \\\\\hline\end{array}$$

Short answer

List of triples: (TWA, 603, Topeka), (TWA, 510, Kansas City), (TWA, 1742, Phoenix), (Pan Am, 7809, Bombay), (Pan Am, 2052, Seattle), (Pan Am, 2025, Anaheim), (Piedmont, 170, Peoria), (Piedmont, 816, Albany), (Piedmont, 717, Atlanta).

Step-by-step solution

Identify Airline Routes from the Table

The table provides several airlines: TWA, Pan Am, and Piedmont. Each airline lists a number of cities they fly to from Chicago, along with the distance (in miles) for each flight. For instance, TWA flies to Topeka, Kansas City, and Phoenix, Pan Am flies to Bombay, Seattle, and Anaheim, and Piedmont flies to Peoria, Albany, and Atlanta.

Make a List of Triples \\(X, Y, Z\\)

For each entry in the table, create a triple where \(X\) is the airline name, \(Y\) is the number of miles, and \(Z\) is the city flew to. For example, the first entry for TWA is Topeka, which corresponds to (TWA, 603, Topeka). Repeat this process for each entry in the table.

Compile All Triples

Combine all the generated triples into a cohesive list. The list will contain data for each airline from the table: 1. (TWA, 603, Topeka) 2. (TWA, 510, Kansas City) 3. (TWA, 1742, Phoenix) 4. (Pan Am, 7809, Bombay) 5. (Pan Am, 2052, Seattle) 6. (Pan Am, 2025, Anaheim) 7. (Piedmont, 170, Peoria) 8. (Piedmont, 816, Albany) 9. (Piedmont, 717, Atlanta)

Key concepts

Relational Database

In the realm of database management, a relational database is the cornerstone for organizing and handling data efficiently. This concept revolves around storing data in structured formats across tables, where each table consists of rows and columns. Each column, also known as a field, represents a data attribute, while each row, or record, holds specific data values within these attributes.

Relational databases employ a standard structured query language (SQL) to easily extract, manipulate, and manage data, which makes them powerful in handling large amounts of data. These databases adopt a relational model, meaning they understand data using relations, aka tables, which are often interlinked using keys like foreign keys and primary keys. Such relationships allow for effective data retrieval across various tables and datasets.

Ensuring data completeness, accuracy, and integrity are essential roles of relational databases. They support multipart transactions and utilize mechanisms such as indexing to accelerate querying processes. Furthermore, relational databases can enforce data integrity constraints, such as unique constraints, which prevent data redundancy and maintain consistency within the database.

Ternary Relations

Ternary relations extend the idea of binary relations to include relationships involving three entities. A ternary relation involves triples of elements denoted as \(X, Y, Z\) and can exist in any context where three objects need to be interrelated as in our airline exercise. In this context, each triple denotes a specific airline, the miles flown, and the destination city.

These relations can be represented abstractly in a set \(R\) such that \(R = \{(X, Y, Z) : \, X \, \text{flies} \, Y \, \text{miles to} \, Z\}\). Here, each element of \(R\) is a distinct triplet where each component provides specific information about the travel resource. The interplay among these three variables facilitates databases or systems to understand more complex interactions than binary relations would allow.

Understanding ternary relations is vital in various fields where complex data interactions are common, offering a versatile way to group and analyze entities with multiple attributes.

Graph Theory

Graph theory serves as a fundamental tool in discrete mathematics used to study the interconnectedness of nodes, often representing networks or pathways. This concept translates real-world problems into mathematical models for better understanding and problem-solving.

In graph theory, data sets can be visualized as graphs composed of vertices (nodes) and edges. Each vertex represents an entity, while each edge embodies a direct relationship or path between two entities. For instance, in our context of airline routes, each city can be a node, and each flight connecting two nodes represents an edge.

This approach is particularly beneficial for solving problems related to shortest routes, network flow, and connectivity of components. Graph theory applications range widely from computer sciences, like social networking algorithms, to logistical planning and even biological networks. By recognizing data patterns and relationships, graph theory offers vital insights for complex problem domains and helps in simplifying intricate data interactions for further analysis and computing.