Prepare for your exams
Get points
Guidelines and tips

Sell on Docsity

Log in Sign up

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Sell on Docsity

Log in Sign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents

20 Points

For each uploaded document

Answer questions

5 Points

For each given answer (max 1 per day)

All the ways to get free points

Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study

Go to the blog

learn about modern mathematics, Summaries of Mathematics

Emilio Aguinaldo College Mathematics

Prof. Gabriel Do Rego

mathematics in modern world, business administration and engineering

Typology: Summaries

2023/2024

Available from 11/11/2024

gabriel-cleofas-do-rego 🇵🇭

1 document

1 / 9

This page cannot be seen from the preview

Don't miss anything!

FUNCTIONS AND RELATIONS

RELATIONS

- A relation is a set of ordered pairs

A relation may have more than 1 output for any

given input.

- The set whose elements are the first coordinates

in the ordered pairs is the domain of the relation.

- The set whose elements are the

second coordinates is the range.

- A = { (1, 1), (2, 3), (2,4)}

- Domain: {( 1, 2)} Range: {(1, 3, 4)}

RELATION IN THE REAL WORLD

- Money won after buying a lotto locket.

- The high temperature on July 1st in New York City.

Depends on the year.

- How many words your friend uses

when answering, “How are you?”

- The number of calories in a fast food hamburger.

- Places you can drive to with 1 gallon left in your

gas tank.

FUNCTION

-

It is a relation in which each element in

the domain is paired with exactly one element

in the range.

- A function can have no more than 1 output for

any given input.

EXAMPLES

- The amount of sodas that come out of a vending

machine. depending how much money you

insert.

- The amount of carbon left in a fossil after so

many years.

- The velocity of an object in freefall after being

dropped so many seconds, excluding air

resistance.

- The height of a person at a given time in their life.

- The intensity of a light as you slide its

dimmer switch.

FUNCTION NOTATION

- The notation f(x) defines a function named

f. This is read as “y is a function of x.” The

letter x represents the input value, or

independent variable. The letter y is

replaced by f(x) and represents the output

value, or dependent variable.

UNARY AND BINARY OPERATIONS

UNARY OPERATIONS

- It involves only one value or accepts one value

or operand.

BINARY OPERATIONA

- It can act on two operands “+” and “ – ”

- It takes two values and include the

operations of addition,

subtraction,

multiplication, division and exponentiation.

PROPERTIES OF TWO BINARY OPERATIONS

CLOSURE OF BINARY OPERATIONS

- The product and the sum of any two real numbers

is also a real number.

Partial preview of the text

Download learn about modern mathematics and more Summaries Mathematics in PDF only on Docsity!

FUNCTIONS AND RELATIONS

RELATIONS

A relation is a set of ordered pairs A relation may have more than 1 output for any given input.
The set whose elements are the first coordinates in the ordered pairs is the domain of the relation.
The set whose elements are the second coordinates is the range.
A = { (1, 1), (2, 3), (2,4)}
Domain: {( 1, 2)} Range: {(1, 3, 4)} RELATION IN THE REAL WORLD
Money won after buying a lotto locket.
The high temperature on July 1st in New York City. Depends on the year.
How many words your friend uses when answering, “How are you?”
The number of calories in a fast food hamburger.
Places you can drive to with 1 gallon left in your gas tank. FUNCTION
It is a relation in which each element in the domain is paired with exactly one element in the range.
A function can have no more than 1 output for any given input.

EXAMPLES

The amount of sodas that come out of a vending machine. depending how much money you insert.
The amount of carbon left in a fossil after so many years.
The velocity of an object in freefall after being dropped so many seconds, excluding air resistance.
The height of a person at a given time in their life.
The intensity of a light as you slide its dimmer switch. FUNCTION NOTATION
The notation f(x) defines a function named f. This is read as “y is a function of x.” The letter x represents the input value, or independent variable. The letter y is replaced by f(x) and represents the output value, or dependent variable. UNARY AND BINARY OPERATIONS UNARY OPERATIONS
It involves only one value or accepts one value or operand. BINARY OPERATIONA
It can act on two operands “+” and “ – ”
It takes two values and include the operations of addition, subtraction, multiplication, division and exponentiation. PROPERTIES OF TWO BINARY OPERATIONS CLOSURE OF BINARY OPERATIONS
The product and the sum of any two real numbers is also a real number.

COMMUTATIVITY OF BINARY OPERATIONS

Addition and multiplication of any two real numbers is commutative as seen in their mathematical symbols: Example: x + y = y + x and x ● y = y ● x ASSOCIATIVITY OF BINARY OPERATIONS
Given any three real numbers you may take any two and perform addition or multiplication as the case maybe and you will end with the same answer. DISTRIBUTIVITY OF BINARY OPERATIONS
Distributivity applies when multiplication is performed on a group of two numbers added or subtracted together. Example: z(x ± y) = zx ± zy IDENTITY ELEMENTS OF BINARY FUNCTIONS
An element of the set of real numbers is an identity element for addition / multiplication if x + e = e + x = x and x ● e = e ● x = x This means that the identity is the number that you add to any real number and the result will be the same real number. INVERSES OF BINARY OPERATIONS

OPERATION ON

FUNCTIONS SUM OF

FUNCTIONS

The sum of two functions is found by adding the two functions together. We can add two functions as: (f + g)(x) = f(x) + g(x) Example: f(x) = 2x2 + 3x – 4 and g(x) = 2x + 3 (f + g)(x) = (2x2 + 3x – 4) + (2x + 3) = 2x2 + 5x – 1 DIFFERENCE OF FUNCTION (f – g)(x) = f(x) – g(x) Example: f(x) = 2x2 + 3x – 4 and g(x) = 2x + 3 (f – g)(x) = (2x2 + 3x – 4) – (2x + 3) = 2x2 + x – 7 PRODUCT OF FUNCTION (f•g)(x) = f(x)•g(x) Example: f(x) = 3x – 5 and g(x) = x (f•g)(x) = (3x – 5)•(x) = 3x2 – 5x QUOTIENT OF FUNCTION (f/g)(x) = f(x)/g(x) Example: f(x) = 3x2 + 4x – 3 and g(x) = x (f/g) (x) = (3x2 + 4x – 3)/x = 3x + 4 – 3/x PROBLEM SOLVING STRATEGIES GEORGE POLYA 1887-
The father of problem-solving
A mathematics educator who strongly believed that the skill of problem solving can be taught

To use the guess-check strategy, one follows these steps:
Continuing the procedure until the correct answer is obtained
Note: Use logical reasoning to minimize the number of trials. STRATEGY 4: ACT IT OUT
Acting out the Problem is a strategy in which people physically act out what is taking place in a word problem.
One may use people or objects exactly as described in the problem, or you might use items that represent the people or objects.
Using this strategy, people visualize and simulate the actions described in the problem. STRATEGY 5: WORK BACKWARDS
The “Work Backward” method works well for problems where a series of operations is done on an unknown number and you’re only given the result.
To use this method, start with the result and apply the operations in reverse order until you find the starting number. PROBLEM SOLVING AND REASONING INDUCTIVE REASONING
It is a type of reasoning that uses specific examples to reach a general conclusion.
The conclusion formed by using inductive reasoning is called a conjecture. Note: A conjecture is an idea that may or may not be correct. COUNTEREXAMPLE
Note: Not all conjectures turn out to be true. You can prove that a conjecture is false by finding one counterexample.
A counterexample to a conjecture is an example for which the conjecture is incorrect. DEDUCTIVE REASONING
It is a type of reasoning that uses general procedures and principles to reach a conclusion. - It is the process of reaching a general conclusion by applying general assumptions, procedures, or principles. SOME GENERALIZED PRINCIPLES - PEMDAS - Property of Real Numbers - Property of Inequality - Postulates - Theorems LOGIC PUZZLE - Logic puzzles can be solved by using deductive reasoning and a chart that enables us to display the given information in a visual manner.

INDUCTIVE VS DEDUCTIVE

Inductive

Reaching conclusions based on a series of observations.
Conjecture may or may not be valid or uncertain. EXAMPLES For the last two years, you gave your dad socks as a Christmas present, and he was happy with them. This year, you have bought your dad more socks and conclude that he is going to be happy with the gift. A stockbroker notices that a company's stock decreased significantly during the summer for the last four years. Therefore, he advises his clients not to invest in that company during the summer. Deductive
Reaching conclusions based on previously known facts.
Conjecture are correct and valid or certain EXAMPLES You know that all doctors hold a doctorate. You meet someone called Dr Jones and conclude that Dr Jones has a doctorate. People from France speak French. You have a new colleague who you know is from France. Therefore, you greet them in French. FORMS OF DEDUCTIVE REASONING

Hypothetical syllogism
Categorical Syllogism HYPOTHETICAL SYLLOGISM

It is a type of deductive reasoning consisting of a conditional major premise, an unconditional minor premise, and an unconditional conclusion. Types of hypothetical syllogism
Modus Ponens
It is a hypothetical syllogism with the form If 𝑝 then 𝑞, and 𝑝, Therefore, 𝑞. where 𝑝 and 𝑞 are distinct statements. EXAMPLE: If Allison vacations in Paris, she will have to win a scholarship. Allison is vacationing in Paris. Therefore, Allison won a scholarship.
- Modus Tollens
- It is a hypothetical syllogism with the form If 𝑝 then 𝑞, and ¬𝑞, Therefore, ¬𝑝. where 𝑝 and 𝑞 are distinct statements. EXAMPLE: If the traffic is bad, Jim will be late to the movie. Jim was not late for the movie. Therefore, the traffic was not bad. CATEGORICAL SYLLOGISM
- It is a form of deductive reasoning wherein a categorical conclusion is based on two categorical premises.
- There are four types of propositions that are used in the syllogism:
  - Positive Universal: “All are...” (A)
  - Negative Universal: “None are...” (E)
  - Positive Existential: “Some are...” (I)
  - Negative Existential: “Some are not...” (O)
- There are three types of propositions will be used to create an argument in the following standard form as defined by:
  - Major premise (universal quantifier)
  - Minor premise (existential quantifier)
  - Conclusion (universal or existential)
- It is also denoted by All 𝑝 are 𝑞. 𝑟 is 𝑝. Therefore, 𝑟 is 𝑞. EXAMPLE: All fish are sea creatures. Every shark is a fish. Therefore, every shark is a sea creature.

Independent Variable (IV)

Is the variable that is systematically manipulated by the investigator Dependent Variable (DV)
Is the variable that the investigator measures to determine the effects of the independent variable Example : A science student wants to know the amount of water needed for the growth of a specific plant. Independent Variable : amount of water exposed to the plant Dependent Variable : growth of the specific plant A scientist studies the impact of Drug A on cancer cells. Independent Variable : use of Drug A Dependent Variable : impact on cancer cells A pathologist studies the reaction of Salmonella bacteria to a new antibiotic developed. Independent Variable : use of the new antibiotic Dependent Variable : reaction of Salmonella Bacteria Descriptive Statistics
It involves the collection and classification of data. Keywords - AVERAGE & PERCENTAGE Example:
- A teacher wishes to determine the percentage of students who passed the examination
- A bowler wants to find his bowling average for the past 10 games Inferential Statistics
- It involves the analysis and interpretation of data. Keywords - PREDICT & ESTIMATE Example:
- A manager would like to predict based on previous years’ sales, the sales performance of a company for the next five years. - A politician would like to estimate, based on an opinion poll, his chance for winning in the upcoming senatorial election. POPULATION AND SAMPLE Population
Set of measurements corresponding to the entire collection of units about which the information is sought; group of objects/subjects about which conclusions are to be drawn. Sample
Set of individuals selected from a population, usually intended to represent the population in a research study Identify the population and the sample. In a recent survey, 614 small business owners in the National Capital Region were asked whether they thought their company’s Facebook presence was valuable. Population : The set of all business owners in the National Capital Region. Sample : The participating 614 business owners in the National Capital Region. SAMPLE SIZE

Data

Are measurements or observations. A data set is a collection of measurements or observations. Datum
Is a single measurement or observation and is commonly called a score or raw score. Qualitative Data
Data that deal with categories or attributes Quantitative Data
Data that deals with numerical values Examples Qualitative Data : (1) Color of skin (2) Courses in Computer Engineering Quantitative Data : (3) Number of units in one semester (4) Grade point average Discrete Data
Data that are obtained by counting Continuous Data
Data that are obtained by measuring EXAMPLES Discrete Data (1) number of students in the classroom (2) number of cars in the parking lot

learn about modern mathematics, Summaries of Mathematics

Related documents

Partial preview of the text

Download learn about modern mathematics and more Summaries Mathematics in PDF only on Docsity!

FUNCTIONS AND RELATIONS

RELATIONS

EXAMPLES

COMMUTATIVITY OF BINARY OPERATIONS

OPERATION ON

FUNCTIONS SUM OF

FUNCTIONS

INDUCTIVE VS DEDUCTIVE