3.3 | INDUCTIVE VS DEDUCTIVE REASONING | MATHEMATICS IN THE MODERN WORLD | ALOPOGS

Summary

This educational video script delves into the concepts of inductive and deductive reasoning in mathematics. It explains that inductive reasoning involves forming general conclusions from specific examples, while deductive reasoning starts with general observations to make specific decisions. The script highlights the importance of both methods in mathematical theorems and proofs, and provides examples to illustrate how each type of reasoning is applied. It emphasizes the certainty of deductive reasoning when premises are correct, and the tentative nature of inductive reasoning, which can lead to hypotheses. The video encourages the use of both reasoning methods cyclically to solve problems effectively.

Takeaways

• 🧠 Inductive reasoning involves reaching a general conclusion by examining specific examples, while deductive reasoning starts with a general theory and makes specific decisions based on that theory.
• 🔍 Mathematical reasoning is crucial for evaluating situations, choosing problem-solving strategies, and understanding when solutions can be applied.
• 📚 Today's formal theorems and proofs in mathematics have their roots in inductive and deductive reasoning.
• 🎓 Deductive reasoning provides correct conclusions if the premises are 100% correct, whereas inductive reasoning's conclusions may not always be valid due to the nature of generalizing from specific instances.
• 🔄 Both types of reasoning are often used together in a cyclical manner, with inductive reasoning used to form hypotheses and deductive reasoning to prove them.
• 🚫 A key pitfall to avoid is mistaking inductive reasoning, which is based on the strength of evidence, for deductive reasoning, which is guaranteed to be true if the premises are correct.
• 🔑 Inductive reasoning is valuable for everyday problem-solving, especially when dealing with partial information, whereas deductive reasoning is more challenging to apply outside of controlled settings.
• 🔍 Examples of inductive reasoning include making assumptions about individuals based on observations of a group, while deductive reasoning involves applying general rules to specific cases.
• 📐 In mathematics, deductive reasoning is often used in proofs, where a conclusion is drawn from accepted premises, ensuring the conclusion's validity.
• 🔢 Inductive reasoning can be applied to number patterns to predict the next term in a sequence, such as identifying even numbers or multiples of a certain number.

Q & A

What is mathematical reasoning?

Mathematical reasoning enables us to use all other mathematical skills, recognizing that mathematics is indispensable, makes sense, and can be understood. It involves evaluating situations, choosing appropriate problem-solving strategies, drawing logical conclusions, and determining when solutions can be applied.

How does mathematical reasoning help in problem-solving?

Mathematical reasoning helps in problem-solving by allowing individuals to reflect on solutions, determine their validity, and appreciate the comprehensive nature and influence of reasoning in mathematics.

What is the difference between inductive and deductive reasoning?

Inductive reasoning involves reaching a general conclusion by examining specific examples, while deductive reasoning starts with general observations and makes specific decisions based on that information.

How do mathematicians use inductive and deductive reasoning in theorems and proofs?

Mathematicians use inductive reasoning to form hypotheses and deductive reasoning to prove ideas. The process involves moving from observation to generalization in inductive reasoning and from theory to experiment to validation in deductive reasoning.

Why is deductive reasoning considered to yield perfect conclusions?

Deductive reasoning is considered to yield perfect conclusions because if all the premises are 100% correct, the conclusions drawn from them are also correct and valid.

What is the limitation of deductive reasoning in non-laboratory or science settings?

Deductive reasoning is harder to use outside of laboratory or science settings because it's often challenging to find a set of fully agreed-upon facts to structure the argument.

How can inductive reasoning be used in everyday problems?

Inductive reasoning is used in everyday problems that deal with partial information about the world, allowing for the development of usable conclusions that may not be universally correct.

What is an example of inductive reasoning provided in the script?

An example of inductive reasoning is: 'John is an excellent swimmer, Jan's family has a swimming pool, therefore we conclude that Jan's sister, Mary, must also be an excellent swimmer.'

Can you provide an example of deductive reasoning from the script?

An example of deductive reasoning is: 'All numbers ending in 0 or 5 are divisible by 5. The number 35 ends with 5, therefore 35 is divisible by 5.'

How can inductive and deductive reasoning be used together to solve problems?

Inductive and deductive reasoning can be used together cyclically, for example, by using induction to come up with a theory and then using deduction to determine if it's actually true.

What is the importance of being cautious when using inductive reasoning?

It's important to be cautious with inductive reasoning because not all conclusions can be accepted as they may not be correct in all circumstances, unlike deductive reasoning, which is guaranteed to be true if the premises are correct.

Outlines

00:00

🔍 Introduction to Inductive and Deductive Reasoning

This paragraph introduces the concepts of inductive and deductive reasoning in mathematics. It emphasizes the importance of mathematical reasoning in understanding and applying mathematical skills. Inductive reasoning is described as reaching general conclusions from specific examples, while deductive reasoning involves making specific decisions based on general observations. The paragraph also discusses how these reasoning methods are fundamental to the creation of mathematical theorems and proofs. The process of inductive reasoning starts with data and moves to general conclusions, whereas deductive reasoning starts with theory and moves to specific confirmations. The validity of conclusions in inductive reasoning may vary, but deductive reasoning provides correct conclusions if the premises are correct. The paragraph concludes by encouraging the use of both reasoning methods to solve problems, often in a cyclical manner.

05:01

🔗 Examples and Applications of Reasoning Methods

The second paragraph provides examples to illustrate the difference between inductive and deductive reasoning. Inductive reasoning examples include assumptions about swimming ability based on family traits, generalizing behavior from a single instance, and inferring the contents of a bag from a few observations. These examples highlight the potential for incorrect conclusions due to the nature of inductive reasoning. Deductive reasoning examples involve applying general rules to specific cases, such as divisibility by 5 and properties of geometric shapes. The paragraph also includes exercises to identify whether a given reasoning process is inductive or deductive, and it discusses the use of reasoning in pattern recognition and problem-solving. The examples demonstrate how inductive reasoning can lead to hypotheses and how deductive reasoning can validate these hypotheses.

10:02

📚 Further Exploration of Deductive Reasoning and Proofs

The final paragraph delves deeper into deductive reasoning with a focus on proofs and logical conclusions. It explains that deductive reasoning involves drawing specific conclusions from general principles or premises. The paragraph provides an example of proving that a quadrilateral is a parallelogram by demonstrating that opposite sides are parallel. It contrasts this with inductive reasoning, which formulates hypotheses and theories from specific instances. The paragraph also discusses the certainty of conclusions in deductive reasoning, provided the premises are true, and the inherent uncertainty in inductive reasoning, which can sometimes yield false conclusions even if all premises are true. The discussion concludes with a reiteration of the importance of using both inductive and deductive reasoning in scientific and mathematical contexts.

Keywords

💡Inductive Reasoning

Inductive reasoning is a method of reasoning where the premises are viewed as supplying some evidence, but not full assurance, of the truth of the conclusion. It involves drawing general conclusions from specific examples. In the context of the video, inductive reasoning is used to form hypotheses, such as observing that all observed swans are white and then hypothesizing that all swans are white. The video script illustrates this with examples like concluding that Mary, who has a swimming pool at home, must also be an excellent swimmer because her brother John is.

💡Deductive Reasoning

Deductive reasoning is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. It is a top-down approach where a conclusion is reached based on general premises. The video explains that deductive reasoning is used to prove ideas, starting with a theory and then moving through observations to validation. An example from the script is the deductive reasoning used to conclude that the number 35 is divisible by 5 because it ends with a 5.

💡Mathematical Reasoning

Mathematical reasoning is the process of acquiring and using mathematical knowledge to solve problems. It involves recognizing the importance of mathematics, understanding its logic, and applying it to evaluate situations and solve problems. The video emphasizes that mathematical reasoning is crucial for understanding and applying mathematical concepts, as it allows students to reflect on solutions and determine their validity.

💡Theorems and Proofs

A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and proven for all cases. A proof is a demonstration that shows that a theorem is true. The video script discusses how both inductive and deductive reasoning are foundational to the development of formal theorems and proofs in mathematics. The process involves starting with observations or theories and systematically working towards a proven conclusion.

💡Generalization

Generalization in reasoning is the process of forming a broad conclusion based on specific instances. It is a key aspect of inductive reasoning where specific observations lead to a general rule or principle. The video uses the example of observing multiple specific instances of good-looking individuals being well-behaved and then generalizing that all good-looking people are well-behaved.

💡Hypothesis

A hypothesis is a proposed explanation for a phenomenon, made on the basis of limited evidence, as a starting point for further investigation. In the video, hypotheses are formed using inductive reasoning, where specific observations lead to the formulation of a tentative explanation that can be tested. For example, if one observes that all coins pulled from a bag are pennies, the hypothesis might be that all coins in the bag are pennies.

💡Observation

Observation is the act of watching, noting, or recording information about the world around us. In the context of reasoning, it is the starting point for inductive reasoning, where specific instances are observed to form general conclusions. The video script mentions observation as the initial step in the inductive reasoning process, such as observing that children in a daycare center are playful.

💡Conclusion

A conclusion is a judgment or decision reached after consideration. In reasoning, it is the final result of a chain of reasoning, whether inductive or deductive. The video explains that conclusions in inductive reasoning may not always be valid, whereas conclusions in deductive reasoning are valid if the premises are correct. For instance, the video uses the example of concluding that all numbers ending in 0 or 5 are divisible by 5.

💡Premise

A premise is a statement or proposition that serves as a basis for an argument or inference. In deductive reasoning, the conclusion follows logically from the premises. The video script uses the term to describe the starting point for deductive reasoning, such as the premise that all squares are rectangles, which leads to the conclusion that all squares have four sides.

💡Pattern

A pattern is a regularity in the world or in reasoning that can be used to make predictions or draw conclusions. In the video, patterns are used in inductive reasoning to identify regularities and make generalizations. For example, if a sequence of numbers follows a certain pattern, such as all being even, one can predict the next number in the sequence by applying the pattern.

Highlights

Mathematical reasoning is essential for understanding and applying mathematics effectively.

Inductive reasoning involves reaching general conclusions from specific examples.

Deductive reasoning starts with general observations to make specific decisions.

Deductive reasoning leads to correct conclusions if the premises are 100% correct.

Inductive reasoning forms hypotheses based on observations and patterns.

Deductive reasoning is used to prove ideas by starting with a theory and moving to confirmation.

Inductive reasoning is used for everyday problems with partial information, leading to usable but not always correct conclusions.

Both inductive and deductive reasoning can be used cyclically to solve problems.

Inductive reasoning may not always yield correct conclusions, unlike deductive reasoning which is certain if premises are true.

Examples of inductive reasoning include making assumptions about individuals based on specific observations.

Examples of deductive reasoning include applying general rules to specific cases to reach a conclusion.

Inductive reasoning is illustrated through examples involving assumptions about groups based on specific instances.

Deductive reasoning examples involve applying general principles to specific scenarios to confirm a theory.

Inductive reasoning can be used to identify patterns in numbers, leading to generalizations about sequences.

Deductive reasoning is used in geometry to prove properties, such as proving a quadrilateral is a parallelogram.

The transcript provides a comprehensive overview of the application and importance of inductive and deductive reasoning in mathematics.

Transcripts

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yes morning to all of you

i will discuss our lesson for today

which is

about inductive and deductive reasoning

mathematical reasoning enables us to use

all other mathematical skills with

mathematical reasoning

we recognize that mathematics is

indispensable

that it makes sense and it can be

understood

we then learn how to evaluate situations

opt for appropriate problem solving

strategies

geological conclusions develop and

describe

solutions and identity when those

solutions can be applied

mathematical prisoners are able to

reflect on solutions to problems

and determine whether or not they make

sense

then students can appreciate the

all-encompassing news and influence of

reasoning

as part of mathematics let us first

define

the inductive reasoning and the

deductive reasoning

inductive reasoning is the process of

reaching a general conclusion

by examining specific examples

today's formal theorems and proofs

originated with these two forms of

reasoning

in inductive reasoning likewise refers

to the process of making generalized

decisions after observing or witnessing

repeated specific instances of something

on the other hand deductive reasoning

refers to the process of taking the

information gathered from

general observations and making specific

decisions

based on that information mathematicians

are still using these types of reasoning

to discover new mathematical theorems

and proofs

deductive starts with theory and passes

through

an observation and then

arrived at the confirmation whereas the

inductive reasoning

starts with data and then conclusions

from inferences

or from observations or experimentations

and arrive at the general accepted

truth or what we call the theory

a conclusion reached by inductive

reasoning

may or may not be valid the conclusions

raised by deductive reasoning are

correct

and valid inductive reasoning is used to

form hypotheses

while deductive reasoning is used to

prove ideas

so another this is the diagram for

inductive reasoning

and for deductive reasoning

so for deduction started from theory

then followed by hypothesis observation

and then confirmation or affirmation of

the theory presented

in the induction it starts from

observation

then go to pattern tentative hypothesis

and then arriving at a theory

or a conjecture what

do mathematicians say about deductive

and inductive reasoning deduction gets

you the perfect conclusion

but only if all your premises are 100

percent correct

deduction moves from theory to

experiment

to validation when induction moves from

observation to generalization to fury

deduction is harder to use outside of

laboratory or science settings

because it's often hard to find a set of

fully agreed upon facts

to structure the argument induction is

used constantly

because it's great tool for everyday

problems

that deal with partial information about

our world

and coming up with usable conclusions

that may not be right in

all cases so therefore in inductions

not all conclusions can be accepted

because those may not be right in all

circumstances

number five be willing to use both types

of reasoning to solve problems

and know that they can often be used

together

cyclically as a pair for example

use induction to come up with a theory

and then use deduction to determine if

it's

actually true the main thing to avoid

with these two is

arguing with the force of deduction that

is guaranteed to be true

while actually using induction the

probability is based on strength of

evidence here are some examples of

inductive reasoning

number one we have here john is an

excellent swimmer

jan's family has a swimming pool then

therefore we conclude that jun's sister

mary must also be an excellent swimmer

so with this example it's up to you if

the conclusion or the conjecture is

acceptable

or accepted to be true number two

elijah is good looking elijah is well

behaved

therefore all good looking are well

behaved so we can see here

that the conclusion is based on the

premises but we can say that the

conclusions

the conclusion is not always true

number three the coin pulled from a bag

is a penny a second coin from the back

is a penny

therefore all the coins in the back are

pennies

number next number children in the

daycare center are playful

children in the daycare center like to

play with legos

therefore playful children like to play

legos

for deductive reasoning we have the

general theory

which says all numbers ending in 0 or 5

are divisible by 5.

the number 35 ends with the 5 so this is

the observation

and your conclusion therefore for

confirmation

therefore 35 is divisible by 5 so this

is true next is all squares are

rectangle

all rectangles have four sides therefore

all squares have four sides

and there are all other examples for

deductive reasoning

like those that are stated in number

three and number four

so when deductive reasoning is stated

first

the theory or the accepted truth the

other statements are accepted also to be

true so we have here some examples of

prisoning

and let us identify whether it is a

deduction or induction

we have all cats have a keen sense of

smell

flappy is a cat so flappy has a keen

sense of

smell so since the statement starts with

all

so it is a general statement therefore

it is a deductive

reasoning number two every time you eat

peanuts

your throat swells up and you can

breathe

this is a symptom of people who are

allergic to peanuts

so you are allergic to peanuts you are

you started from the observation and

then you have the conclusion

that when you eat peanuts you have

or you get an allergy therefore you are

allergic to peanuts

therefore it is an inductive reasoning

and these are also other examples of

reasoning

ray is a football player all football

players

weigh more than 170 pounds ray

weighs more than 170 pounds that is

an inductive started from the specific

statement

and arriving at the general statement

since all squares are rectangles and all

rectangles have four sides

so all spheres have four sides

okay that is a deductive reasoning

if a equals b and b equals c

then a equals c that is a deductive

all cars in this town drive on the right

side of the street

therefore all cars in all towns drive on

the right side of the street

that is an inductive reasoning

so when we apply inductive reasoning

in number pattern we have this 2

4 6 8 10 and what is the next number

if we are asked to find the 19th term so

the la

the following number is

2 n the 19th term because

all these numbers are even numbers and

even numbers are divisible by 2.

so to find the 19th term that is

2 times n 2 times n equals

19 for finding the 19th term times 2

equals 38.

next we have here another pattern

this 6 12 24 so we can see that all

these numbers are multiples of three

so we have the first number three

three times two equals six which is also

the second number

then to find the third number maybe

we can multiply 6 by 2 which is 12

and then we got 24 by multiplying 12 by

2.

then what's the next number after 24

so that is 24 times 2 is 48

and then next number is 48 times 2 which

is 96

and maybe the next number after 96

is 96 times 2 is 192.

another example for deductive reasoning

proof

quad is a parallelogram how can you

prove that

so to find the parallelogram

uh there should be two sides which

are parallel so all sides must be

parallel

so we proved that qd is parallel with ua

and also qu is parallel with d8

therefore if this statements are all

true

then quad is a parallelogram

so that is about inductive

and deductive reasoning so we can

we can see that in inductive reasoning

withdraw general principles from

specific instances

while in deductive reasoning it draws a

specific conclusion from general

principles or premises

a premise is a previous statement or

proposition

from which another is inferred as

follows as a conclusion

unlike inductive reasoning which always

involves uncertainty

the conclusions from deductive inference

are certain provided

the premises are true scientists use

inductive reasoning to formulate

hypotheses and theories

and deductive reasoning when applying

them to specific situations

even if all of the premises are true in

a statement

inductive reasoning may still yield

false

conclusion so those are for deductive

and inductive reasoning

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you