Relation and Functions || MATHEMATICS IN THE MODERN WORLD

Summary

This educational video script delves into the mathematical concepts of relations and functions. It explains the real-life applications of relations, such as familial and professional connections, and then transitions into a mathematical framework where a relation is defined as a subset of a Cartesian product. The script clarifies the difference between relations and functions, emphasizing that functions are a type of relation where each element in the domain corresponds to a unique element in the range. It uses examples and the vertical line test to illustrate these concepts, aiming to provide a clear understanding of how ordered pairs and rules govern these mathematical structures.

Takeaways

• 🔢 A relation in mathematics is a subset of the Cartesian product of two sets and is defined by a rule connecting elements from those sets.
• 👥 Examples of relations in real life include familial ties, marital connections, professional relationships, and shared ethnic backgrounds.
• 📚 The domain of a relation consists of the first elements of the ordered pairs, while the range comprises the second elements.
• 🔍 To determine if an ordered pair belongs to a relation, the script uses the formula x - y / 2, checking if the result is an integer.
• 🎓 The script provides a method to identify elements of a relation by substituting values into the given formula and checking for integer results.
• 📉 A function is a specific type of relation where each element in the domain is connected to exactly one element in the range.
• 📑 The script explains that for a set to be considered a function, no two ordered pairs can have the same x-value with different y-values.
• 📈 The vertical line test is introduced as a method to visually determine if a graph represents a function, ensuring each vertical line intersects the graph at most once.
• 📏 The script clarifies that functions are characterized by unique x-values and can have repeated y-values, but not the other way around.
• 📘 It is emphasized that in a function, the y-values should not be squared or have an exponent greater than one, as this would prevent the graph from passing the vertical line test.

Q & A

What is a relation in the context of mathematics?

In mathematics, a relation is a rule that defines a connection between the elements of two sets. More formally, a relation from set A to set B is a subset of the Cartesian product A * B, consisting of ordered pairs (a, b) where 'a' is an element from set A and 'b' is an element of set B.

How is the domain of a relation defined?

The domain of a relation is the set of all first elements of the ordered pairs in the relation. It represents the 'x' values or inputs in the context of a function.

What does it mean for a relation to have a range?

The range of a relation is the set of all second elements of the ordered pairs. It represents the 'y' values or outputs that correspond to the inputs in the domain.

What is the difference between a relation and a function?

A function is a specific type of relation where each element in the domain is related to exactly one value in the range. This means no two ordered pairs can have the same first element (x-value) but different second elements (y-values).

What is the vertical line test in the context of functions?

The vertical line test is a method used to determine if a graph represents a function. A graph represents a function if every vertical line intersects the graph at most at one point.

How can you determine if an ordered pair is part of a relation defined by the rule x - y/2 is an integer?

To determine if an ordered pair (x, y) is part of the relation where x - y/2 is an integer, you substitute the values of x and y into the equation and check if the result is an integer.

What is the domain and range of the relation defined by the rule x - y/2 is an integer, with set A = {1, 2} and set B = {1, 2, 3}?

The domain of the relation is {1, 2}, as these are the possible values for 'x'. The range is {1, 2, 3}, as these are the possible values for 'y' that can result from the relation's rule.

How do you evaluate a function given the equation y = 2x + 2 at x = 2?

To evaluate the function y = 2x + 2 at x = 2, substitute x with 2 in the equation: y = 2(2) + 2, which simplifies to y = 4 + 2, resulting in y = 6.

What does it mean for a function to be one-to-one?

A function is one-to-one if each element in the domain is mapped to a unique element in the range, and no two different elements in the domain have the same image in the range.

Can you provide an example of a function from the script?

An example of a function from the script is the set of ordered pairs {(1, 3), (2, 6), (3, 9)}. Here, each 'x' value is unique and maps to a single 'y' value.

Outlines

00:00

📘 Introduction to Relations and Functions

The script begins with an introduction to Lesson 3, Chapter 2, focusing on the mathematical language of relations and functions. It explains relations in real-life contexts, such as familial and professional relationships, and contrasts these with the mathematical definition of a relation. In mathematics, a relation is a rule connecting elements of two sets, formally defined as a subset of the Cartesian product of those sets. The lesson further clarifies the concept by using ordered pairs and introduces the terms 'domain' and 'range' to describe the first and second elements of these pairs, respectively. An exercise is presented to identify which ordered pairs belong to a defined relation and to determine the relation's domain and range.

05:03

🔢 Detailed Explanation of Relation Exercise

This section delves into the specifics of a mathematical exercise involving a relation defined between two sets, A and B. The relation R is defined such that an ordered pair (x, y) belongs to R if (x - y) / 2 is an integer. The script guides through the process of evaluating each possible ordered pair from the Cartesian product of A and B to determine if they belong to the relation R. It also discusses the criteria for an ordered pair to be part of R, emphasizing that the result of the operation must be an integer. The solution identifies specific pairs that are elements of R and concludes with the determination of the relation's domain and range.

10:04

📊 Transition to Functions and Their Characteristics

The script transitions from discussing relations to defining functions. A function is described as a special type of relation where each element in the domain is associated with exactly one element in the range. The concept is illustrated with examples and the importance of unique x-values in functions is emphasized. The section also explains that functions can be represented as sets of ordered pairs where no two pairs share the same x-value. The script further clarifies that functions must pass the vertical line test, meaning a vertical line drawn through the graph of a function will intersect the graph at most at one point. Examples are provided to illustrate the concept of functions and non-functions based on this criterion.

15:12

📈 Vertical Line Test and Function Evaluation

This part of the script focuses on the vertical line test as a method to determine if a graph represents a function. It explains that a graph represents a function if every vertical line intersects the graph at most once. The script uses various graphs to demonstrate the application of the vertical line test, identifying which are functions and which are not based on the test. Additionally, the section discusses the evaluation of functions using specific x-values, providing an example of how to calculate the output of a function given an input value.

20:12

🧮 Practical Application of Functions

The final paragraph of the script applies the concept of functions to a practical example, demonstrating how to evaluate a function for a given input. It uses the function Q(x) = x - 2x + 2 and shows the step-by-step process of evaluating this function at x = 2. The explanation includes substituting the value into the function's formula and performing the necessary arithmetic operations to find the output. This practical application reinforces the understanding of functions and their evaluation in mathematical contexts.

Keywords

💡Relation

In the context of the video, 'relation' refers to a connection or association between two entities, such as people related by blood or marriage, or students related to their classmates. Mathematically, a relation is a set of ordered pairs where each pair consists of elements from two sets. The video emphasizes that a relation is defined by a rule that connects elements from one set to another, and it is a subset of the Cartesian product of those sets. For example, the relation R from set A to set B is defined by the rule that an ordered pair (a, b) belongs to R if the difference (a - b) divided by 2 is an integer.

💡Cartesian Product

The Cartesian product is a mathematical operation that combines the elements of two sets to form ordered pairs. In the video, it is mentioned as the basis for defining a relation, where a relation is a subset of the Cartesian product of two sets, A and B. The script uses the Cartesian product to illustrate how ordered pairs are formed and how they are used to represent relations.

💡Domain and Range

Domain and range are fundamental concepts in mathematics, particularly in the study of functions and relations. The domain refers to the set of all possible input values (or the first elements of the ordered pairs), while the range is the set of all possible output values (or the second elements of the ordered pairs). The video script discusses how to identify the domain and range of a relation by examining the ordered pairs that satisfy the relation's rule.

💡Ordered Pairs

Ordered pairs are pairs of numbers or elements from two sets that are written in a specific order, typically (a, b). In the video, ordered pairs are used to represent elements of a relation, where each pair must follow the rule defined by the relation. The script provides examples of ordered pairs that are part of a relation and those that are not, based on the rule that the difference between the elements, when divided by 2, must be an integer.

💡Function

A function is a special type of relation where each element in the domain is associated with exactly one element in the range. The video explains that a function is characterized by the uniqueness of the output for each input, meaning no two ordered pairs can have the same first element (x-value) with different second elements (y-values). This concept is crucial for understanding how functions operate in mathematics and how they differ from general relations.

💡Vertical Line Test

The vertical line test is a graphical method used to determine if a graph represents a function. If any vertical line intersects the graph more than once, then the graph does not represent a function. The video script uses the vertical line test to illustrate which graphs are functions and which are not, emphasizing that a function's graph should only intersect a vertical line at one point.

💡One-to-One Correspondence

One-to-one correspondence is a concept where each element of one set corresponds to exactly one element of another set, and vice versa. In the video, this concept is used to describe the nature of a function, where each input has a unique output. The script explains that a function maintains a one-to-one correspondence between its domain and range.

💡Many-to-One and One-to-Many

These terms describe different types of relationships between sets. 'Many-to-one' means that multiple inputs can lead to a single output, while 'one-to-many' means that a single input can lead to multiple outputs. The video script uses these terms to contrast the properties of functions with other types of relations, noting that a function is a 'many-to-one' relationship where each input corresponds to only one output.

💡Integer

An integer is a whole number that can be positive, negative, or zero. In the video, the concept of integers is used in the definition of a relation, where the rule for the relation involves the difference between elements of an ordered pair being divisible by 2 to result in an integer. This highlights the importance of integers in defining mathematical relations.

💡Rational Number

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. The video script mentions rational numbers in the context of distinguishing them from integers. It explains that a relation's rule may involve conditions that exclude rational numbers (like fractions) to ensure the relation's pairs meet the criteria of being integers.

Highlights

Definition of relation in real-life context and mathematics.

Explanation of blood and marriage relations in society.

Discussion of student-teacher and employer-employee relationships.

Introduction to the concept of a relation in mathematics as a subset of a Cartesian product.

Formal definition of a relation from set A to set B.

Description of ordered pairs in the context of relations.

Explanation of domain and range in relations.

Example problem solving involving identifying elements of a relation.

Use of the formula x - y / 2 to determine if an ordered pair is in a relation.

Solution to the problem of identifying which ordered pairs are in relation R.

Definition of a function as a special type of relation.

Explanation of the one-to-one correspondence in functions.

Description of how a function is represented on a graph using the vertical line test.

Example of evaluating a function using a given equation.

Discussion on the characteristics of a function in terms of ordered pairs.

Explanation of the difference between a function and a non-function using the vertical line test.

Illustration of one-to-many and many-to-one relationships in the context of functions.

Final summary of the key points about functions and relations.

Transcripts

so let's continue for the lesson 3

lesson

3 Chapter 2 lesson 3 topic mathematical

language and symbol the language of

relation and functions Okay let's

start uh what is relation in the real

life situation in the real life problem

so the relations there are various type

of relations in the world for example

two people are considered related by

blood if they share a common ancestor so

that is a relation and related by

marriage If one has a common an sister

with other spouse okay and We also refer

to relationships between

students teachers people who work for

the same employer and you as a students

Uh you have your relation to your

classmate because you are Uh You are a

classmate No you're Schoolmate ' ba and

you are in rolled here in this deas okay

and individual so share a common ethnic

background so that is a relation so in

math no ah in mathematics a relation is

a rule that defines a connection between

the elements of two sets Okay more

formally a relation from the set a to

set B is a subset of the cartesian

product a * b and this means

it ah it consists of ordered pairs A and

B enclosed by a parenthesis Okay take

note enclose by a parenthesis where a is

an element from set A and B is an

element of set B if aare A and B belongs

to the relation we say that a Okay we

just ignore is related to B by this

relation and and the characteristic of

the relation take not ha that the

characteristic of the relation we have

our domain and range ung domain is that

is our x and range that is our y so

domain the set of all first elements

of of a of the ordered pairs so as you

can see meron tayong A and B dito yung a

is our domain and B is

our range Okay so X and Y yan so range

the set of all second elements no B of

the ordered pairs A and B so ito Iyung

range Okay yan no okay and in the

relation so a Relation of a subset no

let's try to let's try to solve this one

Okay let's try to solve this

one let set a is equ to 1 and 2 2 and

set b 1 2 3 So ito yung mga element ng

Set a 1 2 set b 1 2 and 3 and define a

relation

R from a to B as follows given any X and

Y na ordered pair element of a * b so X

and Y element of R means that x - y / 2

is an integer so let's try to solve this

one

Okay we have three questions here state

expressly which order pairs are a times

b and which are in R is 1 relation to 3

is two relation to 3 is two relation to

2 and what are the domain and range of

the relation so let's try to solve this

one Okay so let's using

um let's try to solve okay

so let A1 and 2 set b 1 2 3 and define a

relation from R from a to B as follows

given any X and Y element of a b x and y

element of R means that x Min y is an

integer So take note CL using this

formula no na given is x - y div 2 is an

integer so number one state explicitly

which ordered are in a * b and which are

in R okay a * b so ito yung mga ah value

natin no okay a Tim B so meron tayong

take ha a Tim b 1 and 1 yan 1 and

2 1 and

3 Okay 2 * 1 yan 2 Tim 2 yan and 2 and 3

Okay so yan so let's try to solve using

this formula x - y div 2 so let's try to

solve the first element enclosed by

enclosed by a parenthesis which is 1 and

1 Okay 1 and 1 ito yung

x ito yung

y so 1 and is an element of R so because

a using using substitution method no ito

yung x ' ba ito yung x 1 yung x ung yung

x is 1 ang iung y is 1 so 1 - 1 is 0 so

0 / 2 is zer so take note si 1 and 1 ba

ay yung zero ba is an integer ba Okay

Yes zero is an integer no Lahat ng

negative zer and positive number is an

integer Okay so element yan so check so

1 and two so second element second

element 1 and 2 is not an element of R

Bakit is not an element no bakit hindi

siya element because 1 - 2 / 2 1 - 2 is

-1 no div by 2 1 - 1/2 take note CL Uh -

1/2 is

Uh rational number ' So walang rational

number po no dito sa ating integer So it

means not an element siya Okay so yan

mali 1 and 3 third element na po 1 and

3 next is 1

and 3 So 1 and 3 is an element of R

Bakit element siya So using the ah

formula x - 1 / 2 yung x natin is 1 yung

y natin is 3 So 1 - 3

-2 then div 2 -1 -1 is an integer

check next 2 and 1 so fourth fourth

element is not an element Bakit hindi

siya element ni

R 2 - 1 / 2 is 1 2 - 1 1 and 2 1/2 so

Bawal po ang fraction sa may integer so

not an element of R po Okay so

mali 2 and 2 is an element of R so fifth

element no bakit element siya So 2 - 2 0

/ 2 0 so check check and 2 and 3 is not

an element of R Bakit hindi siya element

no 2 - 3 -1 / 2 - 1/2 1/2 is rational

number po hindi siya integer So it means

x po Okay so yung yung correct answer

lang po no sa

may sa may set of ordered pairs no ito

yung mga elements ng order pairs is si

ang correct lang po dito is na maging

integer po using the x - 1 / 2 1 and 1 1

and 3 2 and two so mali na ng uban no so

malay na yung yung iba okay so bisayan

na ako so

next Okay We're done no in number one

Question number two question is one

relation to

three one relation to 3 is one relation

to 3 yes yes no kasi ah pag i-subsidize

2 relation to 3 2 relation to 3 big no

Kasi - 1/2 yung sagot pag two relation

to 3 So no no

no so next 2 relation to 2 2 and 2 z0

tama Yes Okay so only one and relation

to 3 and 2 relation to 2 not 2 and two

relation to Okay next number 3 What are

the domain and range of r

What

the r and the Range balikan natin sa

ordered pairs balikan natin ito yung mga

domain yung first element sa order pa po

ang mga

domain

Okay Ito po yung mga range yung

naka-check the domain is

okay yung domain dito is

1a 2 1 and 2 lang po no yung range

is ano yung range dito

1 2 and 3 Okay So yan lang po no yung

sagot Okay so range is 1 2 3 and domain

is 1 and 2 Okay Tingan mo lang d sa may

ordered pairs set of ordered pairs so

next

function So what is a function so

function is a relation where each

element in the domain is related to only

one value in the Range by su rule Okay

ba sa function only one value in the

Range by the su rule and the elements of

the domain can be imagined okay as your

input that is your input iung domain to

a machine that applies a rule so that

each input correspond to only one output

so every

Okay there is a corresponding no only

one output in every input so to be

called an a function and a function is

is a set of ordered pairs X and Y such

that no two ordered pairs have the same

x value but different value y values

take not class that yung function po

maging function po yung yung set set of

an order pairs set of of an ordered pair

if and only if ah the

x value is or is or

um not repeated no or unique x value

Okay so for example for example Okay so

This is an example of function Okay so

in a table of values ba using the table

of values we can graph the cartesian

plane as you can see yung mga x variable

po dito yung x value po dito is

nag repeat no walang reputation of

numbers Okay walang reputation of number

so that is a function no Okay lang po na

magrep lang ang mga value ng y values no

the y values Basta hindi lang magrep

Iyung x value okay to be called a ah

function Okay ordered pairs as you can

see ' ba yung -2 -1 0 1 and 2 is very

unique yung mga x values

So it means function yan in the graph

naman dapat using the vertical line test

only one point lang po ang

mag-in bawal na Ang duha point so to be

called a function in the equation naman

no dapat yung y value is walang exponent

na

2 walang squared yung y values y

variable kasi pag mag y s na siya yung

graph niya is naka circle na no or naka

oval ah using the vertical line test mag

intersect na siya into Two points So

hindi na siya ma Okay take not ha basta

equation dapat yung y values is naka

exponent lang ng 1 no para to call

function so let's try to solve this one

Bakit function yung first set of ordered

pet function po no So take note

that Okay take note that the x value

Okay This is our x value this are the x

values Okay ' ba take note dapat yung

function is hindi magrep yung x values

take note walang nagrepeat na x value So

it means function yan Okay lang po

magrep yung y values Basta hindi lang po

si x

walang problema no kahit anong value pa

ni y Basta hindi lang babalik na babalik

si x value si x okay and the set of Uh

set G no set set of an order pa of G 131

4 25 26 37 take note ito yyung x value

natin Okay take note that yung x value

is nagrepeat

So hindi na iyan mga function So it

means not a function No not a function

na iyan okay basta bumabalik na si x

Okay hindi na yan magfunction

Okay 1 3 2 6 3 9 and

n 3n So take note CL no 1 3 26 3 9 So it

means yung kasunod dito is 4

3n so 3

times

Okay wait lang 3 1 2 3 4 3 * 4 is 12

no Okay so yung kasunod dito Class is

mga unique variable na So it means

function na yan Okay using this formula

no naka pattern na

yan Okay next naman ah using the ah

graph No One to many many to one Wait

lang

ha many to

one Okay so ito yung mga graph no Ito po

is

12 one

one and 12 one is still a function po no

one to one is still a function

Okay function po

yan one to one correspondence is

function po so next naman

is Ano yan many to one or One to many So

yung

isa is main to one so that is M to one

no

[Musika]

m to 1

okay to one so many to one

correspondence po ito okay still a

function

po Okay so yung mga value ng x is hindi

bumabalik

no next naman is One to

many Okay One to many is not a

function not a

function not function

Okay so yan function Yan kasi yung x is

bumabalik no repeated values yung x So

what is a vertical line test Okay the

vertical line test is a visual method na

used in mathematics to determine with a

graph represent a function a graph

represent a function if and only if

every vertical line draw to the graph

intersect it at most one So take note ha

dapat ung using the vertical line test

vertical line test is ung ung line na

naka

Uh vertical line no no nakatin doog siya

no nakab barog Okay this Because is in a

function no each input or x values must

correspond to exactly one output or y

value so using the vertical line test

class no' Ito po ung mga graph natin no

let's try to solve this one

using the vertical line test pag

vertical line test only one point lang

po ang mag intersect Okay let's try to

Ito Iyung graph ha na naka curve no

parang wave no so using the vertical

line Test di ba yung

intersect niya is only one point lang po

' ba nag intersect lang siya only one

point so that is basta mag-in in only

one point so that is

function Ito naman yung

isa yung isa ito na line

nak using the vertical line test

Okay using the vertical line test is

only one point lang po ang intersection

niya So function po

ito

next using the vertical L test itong nak

nak u line ur yan only one point lang po

function po yan Ito yung

circle function or

not Two points na po yung intersection

niya means not a function nf na yan next

naman

ito

ito yan function or not so not a

function

kasi inter

only intersection niya is

2

it so next let's try to evaluate y 2x +

yung equation take not basta walang

square y value

function walang exponent naung y values

function

no yung y values not a function yan no

magcc yan circle yung yung graph niyan

ito function

function

function so let's try to evaluate no the

function so Q of x x - 2x + 2 at x = 2

no so using this substitution method

lang now lahat ng x value po dito is

i-subsidize

[Musika]

that would be all no for

the function and

relation okay