Database Management Systems: Sets, Relations, and Keys, Slides of Introduction to Database Management Systems

Download Database Management Systems: Sets, Relations, and Keys and more Slides Introduction to Database Management Systems in PDF only on Docsity!

Database Management Systems Design

On Sets and Relations

• A set S is a collection of objects, where there are

no duplicates

• Examples
  • A = {a, b, c}
  • B = {0, 2, 4, 6, 8}
  • C = {Jose, Pedro, Ana, Luis}

• The objects that are part of a set S are called the

elements of the set.

• Notation:
  • 0 is an element of set B is written as 0 ∈ B.
  • 3 is not an element of set B is written as 3 ∉ B.

Cardinality of Sets (cont.)

• Some examples:

• A = {a,b,c}
  • |A| = 3
• R – set of real numbers
  • |R| = ∞
• E = {0, 2, 4, 6, 8, 10, …}
  • |E| = ∞
• ∅ the empty set
  • | ∅ | = 0

Set notations and equality of Sets

• Enumeration of elements of set S
  • A = {a,b c}
  • E = {0, 2, 4, 6, 8, 10, …}
• Enumeration of the properties of the elements in S
  • E = {x : x is an even integer}
  • E = {x: x ∈ I and x%2=0, where I is the set of integers.}
• Two sets are said to be equal if and if only they both

have the same elements

• A = {a, b, c}, B = {a, b, c}, then A = B
• if C = {a, b, c, d}, then A ≠C
  • Because d ∉ A

Set Union

• Let A and B be two sets. Then, the union of A and

B, denoted by A ∪ B is the set of all elements x

such that either x ∈ A or x ∈ B.

• A ∪ B = {x: x ∈A or x ∈ B}

• Examples:

• A = {10, 20 , 30, 40, 100}, B = {1,2 , 10, 20} then A ∪ B

• C = {Tom, Bob, Pete}, then C ∪ ∅ = C
• For every set A, A ∪ A = A

Set Intersection

• Let A and B be two sets. Then, the intersection of A and B, denoted by A ∩ B is the set of all elements x such that x ∈ A and x ∈ B. - A ∩ B = {x: x ∈A and x ∈ B}
• Examples:
  • A = {10, 20 , 30, 40, 100}, B = {1,2 , 10, 20} then A ∩ B = {10, 20}
  • Y = {red, blue, green, black}, X = {black, white}, then Y ∩ X = {black}
  • E = {1, 2, 3}, M={a, b} then, E ∩ M = ∅
  • C = {Tom, Bob, Pete}, then C ∩ ∅ = ∅
• For every set A, A ∩ A = A
• Sets A and B disjoint if and only if A ∩ B = ∅
  • They have nothing in common

Power Set and Partitions

• Power Set: Given a set A, then the set of all possible

subsets of A is called the power set of A.

• Notation:
• Example:
  • A = {a, b, 1} then = {∅, {a}, {b}, {1}, {a,b}, {a,1}, {b,1}, {a,b,1}}
  • Note: empty set is a subset of every set.
• Partition: A partition ∏ of a nonempty set A is a subset

of such that

• Each set element P ∈ ∏ is not empty
• For D, F ∈ ∏, D ≠ F, it holds that D ∩ F = ∅
• The union of all P ∈ ∏ is equal to A.
• Example: A = {a, b, c}, then ∏= {{a,b}, {c}}. Also ∏ = {{a}, {b}, {c}}. But this is not: M = {{a, b}, {b}, {c}}

A 2 A 2 A 2

Cartesian Products and Relations

• Cartesian product: Given two sets A and B, the

Cartesian product between and A and, denoted

by A x B, is the set of all ordered pairs (a,b) such

a ∈ A and b ∈ B.

• Formally: A x B = {(a,b): a ∈ A and b ∈ B}
• Example: A = {1, 2}, B = {a, b}, then A x B = {(1,a), (1,b),

(2,a), (2,b)}.

• A binary relation R on two sets A and B is a subset

of A x B.

• Example: A = {1, 2}, B = {a, b}, then A x B = {(1,a), (1,b),

(2,a), (2,b)}, and one possible R ⊆ A x B = {(1,a), (2,a)}

The Relational Model of Data

id name age address

123 Bob 19 San Juan

45678 Tim 53 New York

100971 Mary 25 Miami

Representing information about customers

for a store.

Data is viewed in tabular form This is equivalent to a relation

The Relational Model of Data

• Proposed in 1970 by E.F. Codd

• Models a data set as a table

• Each column represent a value
• Each row represents a group of values that represent a relationship. This relationship describes an given entity. Thus every row in a table describes an entity.
• Columns are often called attributes because they describe information about the entity describes in the row.
• Rows are often called records. Docsity.com

Relation Schema

• Describes the name and columns in a relation
• Components of the schema:
  • Relation name
  • Name of each attribute (also called column or field)
  • Domain of each attribute
    • Described by domain name (often type name) and values.
• Example:
  • costumers(id:string, name:string,age:integer,address:string); - Name: costumers - Attributes: id, name, age, address - Domains: strings and integers

Relation Instance

• A set of tuples (records) in which each tuple

has the same number of attributes as in the

schema.

• Example:

• Schema
  • costumers(id:string, name:string,age:integer,address:string);
• Instance of Customer

id name age address

123 Bob 19 San Juan

45678 Tim 53 New York

100971 Mary 25 Miami

Keys on a relation

• superkey – a set of one or more attributes that

uniquely identify a tuple r in a relation R.

• Minimal superkey – a key from which if we remove an

attribute a, it would no longer be a superkey.

• candidate keys – set of all minimal superkeys in a

relation R.

• primary key – a candidate key that the database

designer chooses as the means to identify the

tuples in a relation R.

Examples of keys

• Super keys : {id,name}, {id, name, address}
• candidate key : {id}
• primary key : {id}

id name age address

123 Bob 19 San Juan

45678 Tim 53 New York

1253 Joe 19 Los Angeles

37322 Bob 30 San Juan

100971 Mary 25 Miami