Functions and Relations.pptx

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Functions and Relations

Every function is a relation

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Objectives for today

Relations



Binary relations and properties



Relationship to functions

n-ary relations



Definitions



CS application: Relational DBMS

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Definition of a relation

•

At the very least, a relation is a set of ordered

pairs

•

An ordered pair consist of a

x

and

y-

coordinate

–

A relation

may be viewed as

ordered pairs, mapping

design, table, equation, or written in sentences

•

x

-values are inputs, domain, independent variable

•

y

-values are outputs, range, dependent variable

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Binary Relations

•

Definition:

Let A and B be two sets. A binary

relation from A to B is a subset of A × B.

•

In other words, a binary relation R is a set of

ordered pairs (a

_i

, b

_i

) where a

_i

A and b

∈

_i

B.

∈

•

Notation: We say that

•

a R b if (a,b) R

∈

•

a R b if (a,b) R

∉

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Example: Course Enrollments

• Let’s say that Alice and Bob are taking CS 441. Alice is also taking Math 336. Furthermore, Charlie is taking Art 212 and Business 444. Define a relation R that represents the relationship between people and classes.

• _Solution:

• Let the set P denote people, so P = {Alice, Bob, Charlie}

• Let the set C denote classes, so C = {CS 441, Math 336, Art 212, Business 444)

• By definition R P × C⊆

• From the above statement, we know that

• _{(Alice, CS 441) R}∈

• (Bob, CS 441) R∈

• (Alice, Math 336) R∈

• (Charlie, Art 212) R∈

• (Charlie, Business 444) R∈

• So, R = {(Alice, CS 441), (Bob, CS 441), (Alice, Math 336), (Charlie, Art 212), (Charlie, Business 444)}

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We can also “relate” elements of more than

two sets―n-ary relations

• _Definition:

Let A1, A2, …, An be sets. An n-ary relation on these sets is a subset of A1 × A2 × … × An. The sets A1, A2, …, An are called the domains of the relation, and n is its

degree.

• _Example_:

Let R be the relation on Z × Z × Z consisting of triples (a, b, c) in which a,b,c form an arithmetic progression. That is (a,b,c) R iff there exist some integer k such that ∈ b=a+k and c=a+2k.

• _{What is the degree of this relation?} • _{What are the domains of this relation?}

• _{Are the following tuples (records) in this relation?} • _{(1,3,5) ??}

• _{(2,5,9) ??}

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N-ary relations

•

Solution

•

What is the degree of this relation? 3

•

What are the domains of this relation? Ints,

Ints, Ints

•

Are the following tuples (records) in this

relation?

•

(1,3,5) 3=1+2 and 5= 1+2*2

•

(2,5,9) 5=2+3 but 9 ≠ 2+2*3

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n-ary relations

We can have relations between more than just 2 sets

A binary relation involves 2 sets and can be described by a set of pairs A ternary relation involves 3 sets and can be described by a set of triples …

An n-ary relation involves n sets and can be described by a set of n-tuples

Relations are used to represent computer databases

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n-ary relations n-ary Relations

Note: N is the set of natural numbers {0,1,2,3,…} An example

Note: R could be considered as an extensional representation of the

ternary relation a<b<c, assuming domains are finite and really quite small

The relation has degree 3

The domains of the relation are the set of natural numbers

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n-ary relations n-ary Relations

Note: N is the set of natural numbers {0,1,2,3,…} Z is the set of integers {…,-2,-1,0,1,2,…}

An example

Note: R could be considered as an extensional representation of the

ternary relation a<b<c, assuming domains are finite and really quite small

The relation has degree 4

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Relational databases

Database is made up of records.

Typical operations on a database are

•_{find records that satisfy a given criteria} •_{delete records}

•_{add records} •_{update records}

Some everyday databases

•_{student records} •_{health records} •_{tax information}

•_{telephone directories} •_{banking records}

•_…

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Relational databases The relational data model

Database made up of records, they are n-tuples, made up of fields

Student record might look as follows

(name,metricNo,faculty,gpa)

(Jones,200401986,Arts,4.9) (Lee,200408972,Science,3.6)

(Kuhns,200501728,Humanities,5.0) (Moore,200308327,Science,5.5)

relations (in relDB) also called tables

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Example from the book Attributes: name, metric No, Dept and GPA

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primary key:

An attribute/domain/column is a primary key when the value of this attribute uniquely defines tuples

i.e. no two tuples have the same value for that attribute

Name cannot be a primary key, neither can Dept or GPS metricNo is a primary key

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The current collection of n-tuples (records) in the relation (table) is called the extension of the relation

The permanent aspects of the relation (table) such as the attribute names is called the intention of the relation

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A composite key is a combination of attributes That uniquely define tuples

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Functions

•

Function of x – a relation in which no two

ordered pairs have the same x-value

•

Examples:

(5, 5) and (5, 2)

not a function

(3, 5) and (5, 2)

function

(13, -24) and (13, 76) not a function

(1, 24) and (7, 24)

function

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9/22/20 09:20 PM 1-6 Relations and Functions 20

Function Notation

f(x

) means function of

x

and is read “

f

of

x

.”

f(x

) = 2

x

+ 1 is written in function notation.

The notation

f(

1)

means to replace

x

with

1

resulting in

the function value.

f

(1) = 2

x

+ 1

f

(1) = 2(1) + 1

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Terms related to functions

•

Domain, codomain, range, image and pre-image

•

Domain: If f is a function from A to B , we say that

A is the domain of f and B is the codomain of f.

•

If f(a) = b, we say that b is the image of a and a is

the pre-image of b.

•

The range of f is also the set of all images of

elements of A. Also if f is a function from A to B,

we say that f maps A to B.

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Example 1

•

What is the domain?

{0, 1, 2, 3, 4, 5}

What is the range?

{-5, -4, -3, -2, -1, 0}

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Example 2

Choice One

Choice Two

3

1

0

–1

2

3

2

–1

3

2

3

–2

0

Which mapping represents a function?

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Example 3

A.

B.

Which mapping represents a function?

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Example 4

a. The items in a store to their prices on a certain date

b. Types of fruits to their colors

Which situation represents a function?

There is only one price for each different item on a certain date. The relation from items to price makes it a function.

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Vertical Line Test

•

Vertical Line Test:

a relation is a

function

if a

vertical line drawn through its graph, passes

through only one point.

AKA: “The Pencil Test”

Take a pencil and move it from left to right (

–x

to

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Vertical Line Test

Would this

graph be a

function?

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Vertical Line Test

Would this

graph be a

function?

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Is the following function

discrete or continuous? What

is the Domain? What is the

Range?

Discrete

-7, 1, 5, 7, 8, 10

 

 

 

1, 0, -7, 5, 2, 8

 

 

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discrete or continuous? What

Range?

continuous

8,8

      





6,6

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Range?

continuous

0,45

       

10,70

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Range?

discrete

-7, -5, -3, -1, 1, 3, 5, 7

 

 

 

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One-to-one function

•

If a function does not assign the same value to

two different domain elements we call such a

function

one-to-one or injective

.

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Onto function

•

For some functions the range and codomain

are equal, i.e. every member of the codomain

is the image of some element of the domain.

Such functions are called

onto functions

or

surjections

.

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One-to-one correspondence

•

A function f is a one-to-one correspondence or a

bijection, if it is both one-to-one and onto.

•

Example: Let f be the function from {a, b, c, d} to [1, 2,

3, 4} with f(a) =4, f(b) = 2, f(c) = 1, and f(d) = 3. Is f a

bijection?

•

The function f is one and also onto. It is

one-to-one because no two values in the domain are assigned

the same function value. It is onto because all four

elements of the codomain are images of elements in

the domain. Hence f is a bijection.

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Inverse Functions and Composition of

Functions

•

Inverse function

•

Let f be a one-to-one correspondence from the set A to the

set B. The

inverse function

of f is the function that assigns

to an element b belonging to B the unique element a in A

such that f(a) = b. The inverse function of f is denoted by f

-1

(b) = a when f(a) = b.

•

Note:

f

-1

is just a notation and is not the same as 1/f

•

A one-to-one correspondence is invertible because we can

define an inverse of that function. A function is not

invertible if it is not a one-to-one correspondence, because

the inverse of such a function does not exist.

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Composition of functions

•

Let g be a function from the set A to the set B and let f be

a function from the set B to the set C. The composition of

the functions f and g, denoted by fog is defined as

•

(fog)(a) = f(g(a))

•

To find (fog)(a) we first apply the function g to a to obtain

g(a) and then we apply the function f to the result g(a) to

obtain

•

(fog)(a) = f(g(a))

•

Note that the composition fog cannot be unless the

range of g is a subset of the domain of f.

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• Example:

• Let f and g be the functions from the set of integers to the set of integers defined by

• _{f(x) = 2x + 3 and g(x) = 3x + 2. What is the composition of f and g? What is the}

composition of g and f?

• _{Both the compositions of fog and gof are defined.} • _{(fog)(x) = f(g(x))}

• _{=f(3x + 2) = 2(3x + 2) +3 = 6x + 7.} • _and

• _{(gof)(x) = g(f(x))} • _{= g(2x + 3)}

• _{= 3(2x + 3) + 2 = 6x + 11}

• _{Remarks: The two results show that composition of functions is not commutative.}

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Ceiling and floor functions

•

Two important functions in discrete maths are the

floor

and

ceiling

functions.

•

Assume

x

is a real number:

•

The

floor function

rounds x

down to the closest

integer

less than or equal to x

, and the

ceiling

function

rounds x to the

closest integer greater than

or equal to x

. These functions are often used when

objects are counted. They play an important role in

the analysis of the number of steps used by

procedures to solve problems of a particular size.

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•

The value of the floor function at x is denoted by

Lx˩

.

•

The value of the ceiling function at x is denoted by

Гx˥

.

•

The floor function is also called the

greatest integer

function

and it is often denoted by

[x]

.

•

Examples:

•

These are some values of the floor and ceiling

functions.

•

L1/2˩ =0, Г1/2˥ = 1, L-1/2˩ = -1, Г-1/2˥ = 0, L3.1˩ = 3,

Г3.1˥ = 4, L7˩ = 7, Г7˥ = 7