2.1 Use Inductive Reasoning Objectives 1.To form conjectures through inductive reasoning 2.To disprove a conjecture with a counterexample 3.To avoid fallacies.

2.1 Use Inductive Reasoning2.1 Use Inductive Reasoning

Objectives

1. To form conjectures through inductive reasoning

2. To disprove a conjecture with a counterexample

3. To avoid fallacies of inductive reasoning

Activity: Murder MysteryActivity: Murder Mystery

Once upon a time long, long ago in a far, far away land known as Geometrica there occurred an unspeakable crime. On a dark and dreary night as the Circular family lay sleeping in their soft, round beds and dreaming of their favorite dessert, pi, a violent criminal murdered them. Their neighbor, Mrs. Equi Angular said that she and her husband, Mr. Tri Angular, heard the awful blood curdling screams.

Activity: Murder MysteryActivity: Murder Mystery

So, they sprang from their bed to see what was the matter, and what to their wandering eyes did appear (not eight tiny reindeer) but a strange four-sided figure leaping from the Circular’s upstairs window. Well, the Angulars gave a description of the terrible beast and so did many other Geometrica residents. However, to this day, the mystery remains.

Activity: Murder MysteryActivity: Murder Mystery

Therefore, Detective Pentagonal Walsh of Geometrica’s Most Wanted has asked for your assistance in solving this crime. Below you will find descriptions that tipsters have given the authorities. Your job is to list the suspects from your line-up of twelve that meet each set of criteria.

Activity: Murder MysteryActivity: Murder Mystery

In this activity, you used your observational skills to draw a conclusion about who each tip was. That’s called inductive inductive reasoningreasoning. You then used your conjecture to help you solve the who committed the horrible, disastrous murder.

Example 1Example 1

You’re at school eating lunch. You ingest some air while eating, which causes you to belch. Afterward, you notice a number of students staring at you with disgust. You burp again, and looks of distaste greet your natural bodily function. You have similar experiences over the course of the next couple of days. Finally, you conclude that belching in public is social unacceptable. The process that lead you to this conclusion is called inductive reasoning inductive reasoning.

Inductive ReasoningInductive Reasoning

Inductive reasoningInductive reasoning is the process of observing data, recognizing patterns, and making generalizations based on your observations.

GeneralizationGeneralization

Generalization: Generalization: statement that applies to every member of a group

• Science = hypothesis

• Math = conjectureconjecture

ConjectureConjecture

A conjectureconjecture is a general, unproven statement believed to be true based on investigation or observation

Inductive ReasoningInductive Reasoning

Inductive reasoning can be used to make predictions about the future based on the past or to make conjectures about the past based on the present.

Example 2Example 2

A scientist takes a piece of salt, turns it over a Bunsen burner, and observes that it burns with a yellow flame. She does this with many other pieces of salt, finding they all burn with a yellow flame. She therefore makes the conjectureconjecture: “All salt burns with a yellow flame.”

Inductive ReasoningInductive Reasoning

Inductive ReasoningInductive ReasoningInductive ReasoningInductive Reasoning

Example 3Example 3

Find the 12th number in the following sequence:

1, 1, 2, 3, 5, 8, …

This is called the Fibonacci SequenceFibonacci Sequence!

Example 4Example 4

Numbers such as 3, 4, and 5 are consecutive numbers. Make and test a conjecture about the sum of any three consecutive numbers.

Example 5Example 5

Use the map of Texas provided to formulate a conjecture about the numbering system of Interstate Highways.

Example 5Example 5

Use the map of the US provided to formulate a conjecture about the numbering system of Interstate Highways.

Example 6Example 6

(An allegory) Student A neglected to do his/her homework on numerous occasions. When Student A's mean teacher popped a quiz on the class, Student A failed. After the quiz, Student A had several other HW assignments that he/she also neglected to complete. When test time rolled around, Student A failed the exam . Students B-F behaved in a similar, academically deplorable manner. Use inductive reasoning to make a conjecture about the relationship between homework and test/quiz performance.

Example 7Example 7

Inductive reasoning does not always lead to the truth. What are some famous examples of conjectures that were later discovered to be false?

To Prove or To DisproveTo Prove or To Disprove

In science, experiments are used to prove or disprove an hypothesis.

In math, deductive reasoning is used to prove conjectures and counterexamples are used to disprove them.

CounterexampleCounterexample

A counterexamplecounterexample is a single case in which a conjecture is not true.

Example 8Example 8

On her first road trip, Little Window Watcher Wilma observes a number of vehicles. Each one she observes has four wheels. She conjectures “All vehicles have four wheels.” What is wrong with her conjecture? What counterexample will disprove it?

Conjecture: All vehicles have 4 wheels

COUNTEREXAMPLE

Example 9Example 9

Prove or disprove the following conjecture:

For every integer x, x2 + x + 41 is prime.

For more information on prime numbers, visit http://www.utm.edu/research/primes/.

Example 10Example 10

Kenny makes the following conjecture about the sum of two numbers. Find a counterexample to disprove Kenny’s conjecture.

Conjecture: Conjecture: The sum of two numbers is always greater than the larger number.

Example 11Example 11

Joe has a friend who just happens to be a Native American named Victor. One day Victor gave Joe a CD. The next day Victor decided that he wanted the CD back, and so he confronted Joe. After reluctantly giving the CD back to his friend, Joe made the conjecture: “Victor, like all Native Americans, is an Indian Giver.” What is wrong with his conjecture? What does this example illustrate?

Inductive FallaciesInductive Fallacies

The previous example illustrated an inductive inductive fallacyfallacy, where a reliable conjecture cannot be justifiably made. Joe was guilty of a Hasty Hasty GeneralizationGeneralization, basing a conclusion on too little information. Here are some others:

• Unrepresentative SampleUnrepresentative Sample• False AnalogyFalse Analogy• Slothful InductionSlothful Induction• Fallacy of ExclusionFallacy of Exclusion

Inductive FallaciesInductive Fallacies

As a group, match each inductive fallacy definition with the corresponding example. Be sure to take some notes, as this priceless information is not in your textbook.

AssignmentAssignment

• Inductive Fallacy Examples

• Challenge Problems