# Discrete Mathematics

· The Assessment 1 Context document contains additional information about the concepts and language of sets, functions, sequences, and relations and their application to relational databases.

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Resources

Suggested Resources

The following optional resources are provided to support you in completing the assessment or to provide a helpful context. For additional resources, refer to the Research Resources and Supplemental Resources in the left navigation menu of your courseroom.

Capella Resources

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· Assessment 1 Context.

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Capella Multimedia

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· Functions and Relations | Transcript.

2. This presentation introduces the following topics.

1. Functions.

1. Modulus operator.

1. One to one functions.

1. Onto functions.

1. Composite functions.

1. Sequences.

1. Sigma notation.

1. Relations.

1. Order relations.

Library Resources

The following e-book from the Capella University Library is linked directly in this course:

· Koshy, T. (2004). *Discrete mathematics with applications* . Burlington, MA: Elsevier Academic Press.

. Chapters 2, 3, and 7.

Course Library Guide

A Capella University library guide has been created specifically for your use in this course. You are encouraged to refer to the resources in the MAT-FP2051 – Discrete Mathematics Library Guide to help direct your research.

Bookstore Resources

The resource listed below is relevant to the topics and assessments in this course and is not required. Unless noted otherwise, this resource is available for purchase from the Capella University Bookstore. When searching the bookstore, be sure to look for the Course ID with the specific *–FP* (FlexPath) course designation.

· Johnsonbaugh, R. (2018). *Discrete mathematics *(8th ed.). New York, NY: Pearson.

. Chapter 3, “Functions, Sequences, and Relations,” is particularly useful for your work in this assessment. In addition to focusing on functions, sequences, and relations, this chapter discusses equivalence relations, matrices of relations, and relational databases.

· Assessment Instructions

Imagine your company asks you to organize a relational database containing employee information. A database has *entries* represented by rows and *fields* represented by columns. An entry corresponds to each employee, and each employee has various characteristics depicted by each field. In this database, you will store information for the following six fields for each employee: firstName, lastName, SS#, age, yearsAtCompany, and phoneNumber. Complete the following:

· In general, entries in a relational database can be represented as a set of *n-tuples*, where n is the number of fields. Provide 3 sample entries from 3 employees in the company. Feel free to devise their personal information in each field. Write out this information as a set of 3 6-tuples.

· Since the database has 6 fields and is represented as a set of 6-tuples, it is considered an *n*-ary *relation* where n=6. Let us focus in on two fields of our database: firstName and lastName. Given these two fields, we can define a binary relation, represented by the set of 2-tuples, which includes all entries in our database of these two fields. *Note*: The first entries in the 2-tuples represent elements from the domain, and the second entries in the 2-tuples represent elements from the range. Assume you have populated this database with all employee information. Do you believe this binary relation would be symmetric, transitive, and/or reflexive? Justify your answers. Feel free to use examples to help justify.

· In general, a *function* is a specific type of *relation *such that each element in the domain is related to exactly one element of the range. Given the binary relation composed by the fields firstName and lastName, is this relation a function? Justify your answer.

· In the context of databases, a *key* is a field in the database with the particular characteristic that each value in this field is related to only 1 entry (row) in the database. It is a unique identifier. Given this database, which field would be the best key? Justify your answer.

· Using 2 of the fields from this database, describe a binary relation such that the relation is a function. Define the *domain* and the *range* of this function. Describe this function onto (surjective), one-to-one, (injective), or both (bijective). Justify your answer. Feel free to use an illustration to aid in explanation.

Functions, Relations, and Sets Scoring Guide

CRITERIA | NON-PERFORMANCE | BASIC | PROFICIENT | DISTINGUISHED |

Construct representations of relations or functions using set notation. | Does not construct representations of relations or functions using set notation, or major errors exist or no explanation is provided. | Constructs representations of relations or functions using set notation, but there are errors in explanation or construction. | Constructs representations of relations or functions using set notation. | Constructs representations of relations or functions using set notation and investigates or demonstrates other appropriate representations. |

Apply the concepts of functions, relations, and sets to relational databases. | Does not apply the concepts of functions, relations, and sets to relational databases, or major errors exist or no explanation is provided. | Applies the concepts of functions, relations, and sets to relational databases, but there are errors in explanation. | Applies the concepts of functions, relations, and sets to relational databases. | Applies the concepts of functions, relations, and sets to relational databases; and analyzes there use in this context. |

Construct relations and functions within the context of relational databases. | Does not construct relations and functions within the context of relational databases, or major errors exist or no explanation is provided. | Constructs relations and functions within the context of relational databases, but there are errors in construction or explanation. | Constructs relations and functions within the context of relational databases. | Constructs relations and functions within the context of relational databases and evaluates their use in this context. |

**ASSESSMENT 1 CONTEXT**

All fields of study have a unique language through which those who study and work in that field can communicate. The language of mathematics, through its exquisite elegance and aim to disambiguate, is perhaps the most commonly spoken language in the world.

Relations

Sets are unordered, collections of objects; relationships can be defined between sets of values or objects. Some examples of relations are sequences, functions, or mappings.

As an example, suppose you are given a function *f* on a value *x*, *f*(*x*). Next, suppose the *f*(*x*) is defined as:

*f*(*x*) = 3*x* + 4.

Here, the function *f* is an infinite set of real numbers (the range) that is mapped to or defined by an infinite set of real numbers (the domain) such that for any value *x* in the domain, the value of *f*(*x*) is mapped to or defined by 3*x* + 4. This function tells us that the value 1 is mapped to the value 3(1) + 4, or 7. We can also say that *f*(1) = 7. Relationships between objects or values can be defined in many ways, and you will look at examples of functions, sequences, modulus, hashes, strings, and database applications.

Functions

A function is a special type of relation. A function assigns or maps each value in the domain (usually known as *x*) to exactly one member of the range (usually known as *y*). Functions can be discrete or continuous. An example of a discrete function is the modulus function. This function returns the remainder value resulting from a division operation. For example, 6 mod 5 = 1 and 13 mod 7 = 6.

*f*(*x*) = x mod 31 is a discrete function.

One can describe a function as surjective, injective, and/or bijective.

Sequences

A sequence is a special type of function. For a sequence, the domain is always a set of consecutive integers, such as {1, 2, 3, 4}. The range is a function of that set of integers and is often denoted as *Cn*, where *n* is the *n*th term in the sequence.

Relations and Equivalence Relations

A relation is also a mapping from one set to another, but with fewer constraints than those of functions and sequences. There are different types of relations:

· Binary: A relationship or mapping between two items.

· *n*-ary: A mapping or relationship between multiple items, objects, or numbers.

· Equivalence: A relation that is reflexive, symmetric, and transitive.

Applications: Matrices and Databases

A matrix is an organized method for storing values and processing these values. A matrix can be used to represent a tree or graph, and it can be used to plan or evaluate an algorithm.

A database is a structure for storing data as well as accessing and processing that data. Relations are directly related to databases, as most databases are constructed using relationships between entities and items. These are known as relational databases.

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