### Understanding Relations U and TIn the image provided, we are asked to express the relation \( T \circ U \) using ordered pair notation.#### Diagram Explanation1. **Relation U:** - This is depicted as a directed graph with three nodes labeled \( a \), \( b \), and \( c \). - There are directed arrows indicating the following pairs: - \( (a, b) \) - \( (b, c) \) - \( (c, a) \)2. **Relation T:** - Similarly, this is another directed graph with nodes \( a \), \( b \), and \( c \). - The directed arrows for this relation are: - \( (a, c) \) - \( (b, a) \) - \( (c, b) \)### TaskTo express \( T \circ U \) (the composition of T and U), we need to determine all pairs \( (x, z) \) such that there is an intermediate element \( y \) where both \( (x, y) \) is in U and \( (y, z) \) is in T.#### Steps to Find \( T \circ U \):- Use the given pairs in U to find potential starting points.- Check T for possible connections from these points to create new pairs.#### Example:- From U: \( (a, b) \)- Check T for \( b \): \( (b, a) \)- Resulting pair for \( T \circ U \): \( (a, a) \)Continue similar steps for all combinations to form the full set of ordered pairs.### Final Expression for \( T \circ U \)\[ T \circ U = \{ (a, a), (b, b), (c, c) \} \]This is derived by checking all possible transitions from U to T as described above. Remember, the key is finding intermediate \( y \) such that the transition from \( x \to y \to z \) holds true in the context of the graphs provided.

To express the composition of relations T and U (T o U) in ordered pair notation, we need to…

Answered: Given the relations U and T below, use…

Given the relations U and T below, use ordered pair notation to express the relation To U. ToU = {Ex: (a, b), (b, c) b U T с

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Concepts in Designing Database

A database design is the process of data organization based on a database model. The process deals with identifying what data should be stored in a database and how data elements relate to each other.

Entity Relationship Diagram

Complex real-world applications call for large volumes of data. Therefore, it is necessary to build a great database to store data safely and coherently. The ER data model aids in the process of database design. It helps outline the structure of an organization’s database by understanding the real-world interactions of objects related to the data. For example, if a school is tasked to store student information, then analyzing the correlation between the students, subjects, and teachers would help identify how the data needs to be stored.

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### Understanding Relations U and T

In the image provided, we are asked to express the relation \( T \circ U \) using ordered pair notation.

#### Diagram Explanation

1. **Relation U:**
- This is depicted as a directed graph with three nodes labeled \( a \), \( b \), and \( c \).
- There are directed arrows indicating the following pairs:
- \( (a, b) \)
- \( (b, c) \)
- \( (c, a) \)

2. **Relation T:**
- Similarly, this is another directed graph with nodes \( a \), \( b \), and \( c \).
- The directed arrows for this relation are:
- \( (a, c) \)
- \( (b, a) \)
- \( (c, b) \)

### Task

To express \( T \circ U \) (the composition of T and U), we need to determine all pairs \( (x, z) \) such that there is an intermediate element \( y \) where both \( (x, y) \) is in U and \( (y, z) \) is in T.

#### Steps to Find \( T \circ U \):

- Use the given pairs in U to find potential starting points.
- Check T for possible connections from these points to create new pairs.

#### Example:

- From U: \( (a, b) \)
- Check T for \( b \): \( (b, a) \)
- Resulting pair for \( T \circ U \): \( (a, a) \)

Continue similar steps for all combinations to form the full set of ordered pairs.

### Final Expression for \( T \circ U \)

\[ T \circ U = \{ (a, a), (b, b), (c, c) \} \]

This is derived by checking all possible transitions from U to T as described above. Remember, the key is finding intermediate \( y \) such that the transition from \( x \to y \to z \) holds true in the context of the graphs provided.

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