induction and deduction “come reason with me”. what is the difference between the thinking...

Induction and Deduction

“Come reason with me”

What is the difference between the thinking processes described below?• Adham: I've noticed previously

that every time I kick a ball up, it comes back down, so I guess this next time when I kick it up, it will come back down, too.

• Rizik: That's Newton's Law. Everything that goes up must come down. And so, if you kick the ball up, it must come down.

http://www2.sjsu.edu/depts/itl/graphics/induc/ind-ded.html

Today’s Learning Goals:

• To differentiate between inductive and deductive reasoning.

• To create examples of deductive and inductive reasoning.

• To identify examples of deductive reasoning.

• To identify examples of inductive reasoning.

We are learning this because

• We want to solve logic puzzles.• We would like to make video games.• We want to learn logic.• We want to be able to do mathematical proofs.• We like to figure out patterns.• We want to make good financial decisions.• All of the above

“Cogito ergo Sum” - Rene Descartes

http://www-history.mcs.st-and.ac.uk/~history/PictDisplay/Descartes.html

“Je pense, donc je suis”

Select the correct biker:

In the Tour de France bicycle race, Franz led Henry by 3 miles, while Javier lagged behind Eduardo by 2 miles. Eduardo was ahead of Henry by 1 mile. Halfway through the race, Franz and Eduardo exchanged places (they remained the same distance apart as they were before); then Javier pulled even with Franz. In the last position change before the finish line, Javier dropped back by one mile and Henry passed Eduardo. Who finished third?

a) Franz b) Javier c) Henry d) Eduardo

http://72.14.203.104/search?q=cache:565WMk3m33QJ:www.utexas.edu/student/utlc/ts/tsi/tsi_math/handouts/2571indu_dedu.pdf+%22inductive+and+deductive+reasoning%22&hl=en&gl=us&ct=clnk&cd=13

Solution The first sentence establishes the position of each cyclist in

relation to only one other cyclist (lead cyclist on the right): Henry <-------- 3 miles --------> Franz

Javier <---- 2 miles ----> Eduardo

The second sentence allows the positions of all the cyclists to be linked this way:

Henry <-------- 3 miles --------> Franz,

Javier <-- 1 mile --> Henry <-- 1 mile --> Eduardo

Solution, continuedThe entire problem gives distances in 1 mile increments. Therefore, it is convenient to continue to represent the cyclists’ positions with a horizontal scale where each position is 1 mile from the position on either side of it. Because Eduardo and Franz are separated by 2 miles, the series of dashes represents an empty slot that is one mile ahead of Eduardo and one mile behind Franz:

Javier Henry Eduardo ---- Franz

Solution, completedHalfway through the race, Franz and Eduardo exchange their positions,

Javier Henry Franz ---- Eduardo

and then Javier pulls even with Franz, so the new order is as follows:

Henry Franz/Javier ---- Eduardo

Just before the end of the race, Javier drops back by one mile. This results in:

Javier/ Henry Franz ---- Eduardo

Finally, Henry passes Eduardo, making the final order: Javier Franz ---- Eduardo Henry

So Franz finished third. Therefore, the answer is (a).

In a small hospital, there was only 1 room left, which had space for 2 patients. The nurses were instructed to put patients in the remaining room according to these rules:

1. A rubella patient cannot be put with a tuberculosis patient, unless they are both female.

2. If a rubella patient is male, then he can be put with a female emphysema patient.

3. A cancer patient cannot be put with an emphysema patient if one of them is male and the other female.

4. A tuberculosis patient cannot be put with an emphysema patient if they are the same sex.

Which of the following patients could be put in the room if a female emphysema patient was already accepted into the room?

a) female rubella patient b) female tuberculosis patient

c) male cancer patient d) male tuberculosis patient

SolutionThe first rule says that unless a rubella patient and a tuberculosis patient are both female, they cannot be put together. So the allowable combination is:

rubella(F) and tuberculosis(F)

The second rule states that if a rubella patient is male, then he can be put with a female emphysema patient. Hence, another allowable combination is:

rubella(M) and emphysema(F)

The third rule demands that a cancer patient cannot be put in the same room with an emphysema patient unless they are both of the same sex. This gives two more allowable combinations:

cancer(M) and emphysema(M)

cancer(F) and emphysema(F)

Solution, continuedThe last rule orders that a tuberculosis patient should only be

placed with an emphysema patient if they are of opposite sexes. Again, this gives two more allowable combinations:

tuberculosis(M) and emphysema(F)

tuberculosis(F) and emphysema(M)

Now, find an answer choice that is compatible with the patient who is already in the room– a female with emphysema. Rule two allows a male rubella patient, but that is not one of the answer choices. Rule three allows a female cancer patient, but that is not an answer choice either. The last rule allows a female emphysema patient to room with a male tuberculosis patient, so the answer is d)

Deductive Reasoning

Each of the preceding problems was an example of

Deduction

http://www.socialresearchmethods.net/kb/dedind.htm

Deductive Reasoning


Deductive reasoning works from the more general to the more specific. Sometimes this is informally called a "top-down" approach. We might begin with thinking up a theory about our topic of interest. We then narrow that down into more specific hypotheses that we can test. We narrow down even further when we collect observations to address the hypotheses. This ultimately leads us to be able to test the hypotheses with specific data -- a confirmation (or not) of our original theories.

Which comes next?

A DCB


What is the missing number?

19, 23, ____, 31, 37

A) 26 B) 27 C) 21 D) 29


Solution19, 23, ____, 31, 37

Possible pattern: Add 4, ?, add 6, middle could be add 5, this would give us 28, but 28 is not a solution

A: 19, 23, 26, 31, 37 would be add 4, add 3, add 5, add 6. No pattern

B: 19, 23, 27, 31, 37 would be add 4, add 4, add 4, add 6. No

C: 19, 23, 21, 31, 37 would be add 4, subtract 2, add 10, add 6. NoD: 19, 23, 29, 31, 37 would be add 4, add 6, add 2, add 6. No

Solution, continued19, 23, ____, 31, 37

Get outside your box.

• What else could the pattern represent besides relationships between the numbers?

• What kinds of numbers are each of the numbers in the pattern?

• Are any of the responses prime numbers?

Which comes next?

A B C D


Solution

The first figure gives a small black circle with a bracket under it. Careful study of the sequence shows that the circle is always getting larger, so (a) cannot be the answer, because this circle is too small.

Pay attention to the brackets, which alternate from top to bottom and appear in every other figure. With this in mind, you can assume that the next figure should not have a bracket, because the last one shown in the sequence did. Therefore, (b) can be disregarded as an answer.

You can conclude that the next circle should be light, and not dark, because a dark circle appears every third figure only. This leaves choices (c) and (d).

Solution, continued

This sequence seems to have a circular pattern changing from a dark circle to a light circle, then to two circles, and finally back to a dark circle. Since the last figure was a light circle, the next figure should have two circles, and thus (c) is the answer.

Inductive Reasoning

Each of the preceding problems was an example of

Induction


Inductive ReasoningInductive reasoning works the other way, moving from specific observations to broader generalizations and theories. Informally, we sometimes call this a "bottom up" approach (please note that it's "bottom up" and not "bottoms up" which is the kind of thing the bartender says to customers when he's trying to close for the night!). In inductive reasoning, we begin with specific observations and measures, begin to detect patterns and regularities, formulate some tentative hypotheses that we can explore, and finally end up developing some general conclusions or theories.


Let’s try some more! Two women, Anissa and Crystin, and two men, Brett and

Donald, are athletes. One is a runner, a second is a skater, a third is a gymnast, and a fourth is a tennis player. One day they were seated around a square table.

1. The runner sat on Anissa’s left.

2. The gymnast sat across from Brett.

3. Crystin and Donald sat next to each other.

4. A woman sat on the skater’s left.

Who is the tennis player?

a) Anissa b) Brett c) Crystin d) Donald

Solution

Runner Skater Gymnist Tennis player

Anissa

Brett

Crystin

Donald

Solution

Runner Skater Gymnist Tennis player

Anissa ___ ___ ___ X

Brett X ___ ___ ___

Crystin ___ ___ X ___

Donald ___ X ___ ___

AT DS

CG

BR

What is the next number in the sequence?

-4, 6, 22, 44, 72, ...?

a) 96 b) 100 c) 106 d) 111

Solution

-4, 6, 22, 44, 72, ...?

Possible Pattern: Add 10, Add 16, add 22, add 28…

So we are adding 6 more each time then we did the time before it. So we need to 28 + 6 to 72. This gives us 72 + 34 = 106. So the answer is c.

What a logic puzzle isn’t?

Crossword puzzle

Word search

Rubber chicken

What a Logic Puzzle is”

A logic puzzle IS (usually) a strategically solved mind expanding questionnaire

sometimes solved using a grid mapping out room in which to sort and interpret given

data.

In "Human" English, that means:

A logic puzzle is often a word problem, in which you use a special organized table to

chart out the answer.

Gentlemen and Ties

There are 3 gentlemen in a meeting: Mr. Yellow, Mr. Green and Mr. Brown. They are wearing yellow, green and brown ties. Mr. Yellow says: "Did you notice that the color of our ties are different from our names?" The person who is wearing the green tie says, "Yes, you are right!"

Do you know who is wearing what color of tie?

http://www.freepuzzles.com/puzzles/PuzzlePage.asp?PuzzleNumber=Logic001&CategoryID=2

Gentlemen and Ties

Mr. Yellow Mr. Green Mr. Brown

Yellow _________

Green _________

Brown _________

Names

Ties

Triplet BrothersThere are 3 triplet brothers. They look identical. The oldest is John, he always tells the truth. The second is Jack, he always tells a lie. The third is Joe, he either tells the truth or a lie.

Jimmie Dean went to visit them one day. He was wondering who was who. So he asked each person a question.

He asked the one who was sitting on the left: "Who is the guy sitting in the middle?". The answer was "He is John."He asked the one who was sitting in the middle: "What is your name?". The answer was "I am Joe."He asked the one who was sitting on the right: "What is the guy sitting in the middle?". The answer was "He is Jack."

Jimmie Dean got really confused. Basically, he asked 3 of the same questions, but he got 3 different answers. Would you find out who is who for Jimmie?

Triplet Brothers

left center right

John

Jack

Joe

How old are they? The investigator asked Jaclyn about her children. Jaclyn said "I have 3 daughters, Alice, Betty, and Cindy. The product of their ages is 36. The sum of their ages is the same as the street number of our next door neighbor."

The investigator went next door and came back and said: "Still not enough information". Jaclyn said: "Oh, I forgot to tell you that my oldest daughter is now in school".

The investigator found out the ages of her daughters immediately. Do you know their ages?

Benchmark Examples:

Choose the statement that has the correct type of reasoning identified.

A Inductive Reasoning: Using the order of operations to solve an equation.

B Inductive Reasoning: Predicting your next birthday gift, based on your last 5 birthday gifts.

C Deductive Reasoning: Predicting the next letter in a letter series

D Deductive Reasoning: Determining that you will eat two hamburgers for lunch today, because you have eated two hamburgers every day this week.

Other Problems

Which is an example of inductive reasoning?

A A student solves for the area of a parallelogram by using the formula A = bh

B A baseball player uses the formula for finding batting average to calculate the batting average for every Dodgers player.C A student finds the next number in a pattern of numbersD A student uses the order of operations to simplify an algebraic expression.

Another One

Which of the following is an example of deductive reasoning?

A A magician notices the pattern in a group of shapes and guesses the next shape.

B Sister Maria went to church every morning this week, so you figure that she goes to church every morning.

C Monica uses properties of algebra to solve an equation.

D Maurice counts the number of people who walk by his house on Monday and finds that there were 25. On Tuesday there were 30. On Wednesday there were 35. Maurice thinks that on Thursday there will be 40 people who walk by his house.

What kind of reasoning is illustrated by each example

1. Socrates is a man. All men have minds. Therefore, Socrates is a man.

2. I see the pattern 4, 8, 12, 16, 20. I guess that the next two numbers are 24 and 28.

3. I saw 15 purple roses and no other kind. I conclude that all roses are purple.

4. All rational numbers can be written as a fraction. 3.5 is a rational number. Therefore 3.5 can be written as a fraction.

Inductive Example

Example

Environmental Studies: desert elephants live in the Namib desert. (This is specific: desert elephants, not all elephants.

Rule

Elephants adapt to their environment. (This is general: all elephants.)

Conclusion

Desert elephants are well adapted to the

Namib desert.

http://www.edsnet.na/Resources/TBCM/TBCM13/M13p11-12.htm

Inductive Example:

Example

Natural Science and Health education: Crisps are high in fat. (This is specific: crisps, not all foods.)

Rule

Foods that are high in fat are not very health. (This is general: all foods that are high in fat.)

Conclusion

Crisps are not very nutritious


Beware!

We can say therefore when we draw conclusion. And, as with deductive reasoning, we must be careful how we draw conclusions-in this case, how we apply examples to rules. if the rules can be applied in all cases, our rule is probably correct. But there can be exceptions . Even though desert elephants adapt to their environment in general, there might be reasons why certain desert elephants cannot adapt to the Namibian Desert in particular.


Numerical Example:

Find the next three terms in the sequence given below:

1, 2, 4, …

One response: 7, 11, 16

Another response: 8, 16, 32

Conclusions• Inductive reasoning:

1. Conclusion based on several past observations

2. Conclusion is probably true, but not necessarily true

• Deductive Reasoning:1. Conclusion based on accepted statements

(definitions, postulates, previous theorems, corollaries, and given information)

2. Conclusion must be true if hypotheses are true

Fred Smith Campaign

Fred Smith, Freddy Joe, Tom Jones, John Doe, and Bob Bobhason each ran for a

different public office, meaning each one ran a different ad. These candidates were so

dumb that they thought they were running against each other. Each one ran a different ad. Which person ran for which public office

and who ran what advertisement.

http://library.thinkquest.org/04oct/00723/answer1.htm

The Public Offices:• Commissioner of Truth and Justice, •Evil Dictator of the Universe,•Person That Exists, •Lowly Bug, •President of Anyone Who Is A Lowly Bug.

You know these facts.

The ads were: •"John Doe Stinks!" •"Don't Vote Bob!" •"Freddy J is Evil!" •"Tom is Cool!" and •"Why Vote Smith: The Evil Monster of Fred?"


1. Any self-promoting ad was made by the person it describes, also, any non self-promoting ad was made not by the person it describes.

2. Fred Smith ran for an office that contains the words "Lowly Bug"

3. The person who ran the ad "John Doe Stinks" ran for Commissioner of Truth and Justice.

4. John Doe and the person who ran for Evil Dictator of the Universe met up with Tom Jones to argue about senseless stuff.

5. The person who ran for "Person that Exists" has fewer letters in their last name than 7.

6. Anybody whose first or last name contains a "J" did not run for Commissioner of Truth and Justice.

7. The person who ran the ad, "Why Vote Smith, The Evil Monster of Fred" ran for "President of Anyone who is a Lowly Bug.


induction and deduction “come reason with me”. what is the difference between the thinking...

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